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Course: Algebra 1   >   Unit 14

• Solve by completing the square: Integer solutions
• Solve by completing the square: Non-integer solutions
• Solve equations by completing the square
• Worked example: completing the square (leading coefficient ≠ 1)
• Completing the square
• Solving quadratics by completing the square: no solution
• Proof of the quadratic formula

Solving quadratics by completing the square

• Completing the square review
• Quadratic formula proof review

What you should be familiar with before taking this lesson

• Solving quadratic equations by taking the square root
• Solving quadratic equations by factoring

What you will learn in this lesson

Solving quadratic equations by completing the square, what happened here, how to complete the square.

• The coefficient of x ‍   , which we know to be 6 ‍   , should be equal to 2 a ‍   . This means that a = 3 ‍   .
• The constant number we need to add is equal to a 2 ‍   , which is 3 2 = 9 ‍   .
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  
• (Choice A)   b 2 ‍   A b 2 ‍  
• (Choice B)   ( b 2 ) 2 ‍   B ( b 2 ) 2 ‍  
• (Choice C)   ( 2 b ) 2 ‍   C ( 2 b ) 2 ‍  
• (Choice D)   b 2 ‍   D b 2 ‍  

Solving equations one more time

• (Choice A)   x = 21 − 4 ‍   and − 21 − 4 ‍   A x = 21 − 4 ‍   and − 21 − 4 ‍  
• (Choice B)   x = 69 − 8 ‍   and − 69 − 8 ‍   B x = 69 − 8 ‍   and − 69 − 8 ‍  
• (Choice C)   x = 21 + 4 ‍   and − 21 + 4 ‍   C x = 21 + 4 ‍   and − 21 + 4 ‍  
• (Choice D)   x = 69 + 8 ‍   and − 69 + 8 ‍   D x = 69 + 8 ‍   and − 69 + 8 ‍  
• (Choice A)   x = 1 2 + 3 4 ‍   and − 1 2 + 3 4 ‍   A x = 1 2 + 3 4 ‍   and − 1 2 + 3 4 ‍  
• (Choice B)   x = 2 + 3 2 ‍   and − 2 + 3 2 ‍   B x = 2 + 3 2 ‍   and − 2 + 3 2 ‍  
• (Choice C)   x = 1 2 − 3 4 ‍   and − 1 2 − 3 4 ‍   C x = 1 2 − 3 4 ‍   and − 1 2 − 3 4 ‍  
• (Choice D)   x = 2 − 3 2 ‍   and − 2 − 3 2 ‍   D x = 2 − 3 2 ‍   and − 2 − 3 2 ‍  

Arranging the equation before completing the square

Rule 1: separate the variable terms from the constant term, rule 2: make sure the coefficient of x 2 ‍   is equal to 1 ‍   ..

• (Choice A)   x = 7 − 5 2 ‍   and − 7 − 5 2 ‍   A x = 7 − 5 2 ‍   and − 7 − 5 2 ‍  
• (Choice B)   x = 10 − 5 ‍   and − 10 − 5 ‍   B x = 10 − 5 ‍   and − 10 − 5 ‍  
• (Choice C)   x = 7 + 5 2 ‍   and − 7 + 5 2 ‍   C x = 7 + 5 2 ‍   and − 7 + 5 2 ‍  
• (Choice D)   x = 10 + 5 ‍   and − 10 + 5 ‍   D x = 10 + 5 ‍   and − 10 + 5 ‍  

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• 10.2 Solve Quadratic Equations by Completing the Square
• Introduction
• 1.1 Introduction to Whole Numbers
• 1.2 Use the Language of Algebra
• 1.3 Add and Subtract Integers
• 1.4 Multiply and Divide Integers
• 1.5 Visualize Fractions
• 1.6 Add and Subtract Fractions
• 1.7 Decimals
• 1.8 The Real Numbers
• 1.9 Properties of Real Numbers
• 1.10 Systems of Measurement
• Key Concepts
• Review Exercises
• Practice Test
• 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
• 2.2 Solve Equations using the Division and Multiplication Properties of Equality
• 2.3 Solve Equations with Variables and Constants on Both Sides
• 2.4 Use a General Strategy to Solve Linear Equations
• 2.5 Solve Equations with Fractions or Decimals
• 2.6 Solve a Formula for a Specific Variable
• 2.7 Solve Linear Inequalities
• 3.1 Use a Problem-Solving Strategy
• 3.2 Solve Percent Applications
• 3.3 Solve Mixture Applications
• 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
• 3.5 Solve Uniform Motion Applications
• 3.6 Solve Applications with Linear Inequalities
• 4.1 Use the Rectangular Coordinate System
• 4.2 Graph Linear Equations in Two Variables
• 4.3 Graph with Intercepts
• 4.4 Understand Slope of a Line
• 4.5 Use the Slope–Intercept Form of an Equation of a Line
• 4.6 Find the Equation of a Line
• 4.7 Graphs of Linear Inequalities
• 5.1 Solve Systems of Equations by Graphing
• 5.2 Solve Systems of Equations by Substitution
• 5.3 Solve Systems of Equations by Elimination
• 5.4 Solve Applications with Systems of Equations
• 5.5 Solve Mixture Applications with Systems of Equations
• 5.6 Graphing Systems of Linear Inequalities
• 6.1 Add and Subtract Polynomials
• 6.2 Use Multiplication Properties of Exponents
• 6.3 Multiply Polynomials
• 6.4 Special Products
• 6.5 Divide Monomials
• 6.6 Divide Polynomials
• 6.7 Integer Exponents and Scientific Notation
• 7.1 Greatest Common Factor and Factor by Grouping
• 7.2 Factor Quadratic Trinomials with Leading Coefficient 1
• 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
• 7.4 Factor Special Products
• 7.5 General Strategy for Factoring Polynomials
• 7.6 Quadratic Equations
• 8.1 Simplify Rational Expressions
• 8.2 Multiply and Divide Rational Expressions
• 8.3 Add and Subtract Rational Expressions with a Common Denominator
• 8.4 Add and Subtract Rational Expressions with Unlike Denominators
• 8.5 Simplify Complex Rational Expressions
• 8.6 Solve Rational Equations
• 8.7 Solve Proportion and Similar Figure Applications
• 8.8 Solve Uniform Motion and Work Applications
• 8.9 Use Direct and Inverse Variation
• 9.1 Simplify and Use Square Roots
• 9.2 Simplify Square Roots
• 9.3 Add and Subtract Square Roots
• 9.4 Multiply Square Roots
• 9.5 Divide Square Roots
• 9.6 Solve Equations with Square Roots
• 9.7 Higher Roots
• 9.8 Rational Exponents
• 10.1 Solve Quadratic Equations Using the Square Root Property
• 10.3 Solve Quadratic Equations Using the Quadratic Formula
• 10.4 Solve Applications Modeled by Quadratic Equations
• 10.5 Graphing Quadratic Equations

Learning Objectives

By the end of this section, you will be able to:

• Complete the square of a binomial expression
• Solve quadratic equations of the form x 2 + b x + c = 0 x 2 + b x + c = 0 by completing the square
• Solve quadratic equations of the form a x 2 + b x + c = 0 a x 2 + b x + c = 0 by completing the square

Be Prepared 10.2

Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material.

• Simplify ( x + 12 ) 2 ( x + 12 ) 2 . If you missed this problem, review Example 6.47 .
• Factor y 2 − 18 y + 81 y 2 − 18 y + 81 . If you missed this problem, review Example 7.42 .
• Factor 5 n 2 + 40 n + 80 5 n 2 + 40 n + 80 . If you missed this problem, review Example 7.46 .

So far, we have solved quadratic equations by factoring and using the Square Root Property. In this section, we will solve quadratic equations by a process called ‘completing the square.’

Complete The Square of a Binomial Expression

In the last section, we were able to use the Square Root Property to solve the equation ( y − 7 ) 2 = 12 ( y − 7 ) 2 = 12 because the left side was a perfect square.

We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form ( x − k ) 2 ( x − k ) 2 in order to use the square root property.

What happens if the variable is not part of a perfect square? Can we use algebra to make a perfect square?

Let’s study the binomial square pattern we have used many times. We will look at two examples.

Binomial Squares Pattern

If a , b a , b are real numbers,

We can use this pattern to “make” a perfect square.

We will start with the expression x 2 + 6 x x 2 + 6 x . Since there is a plus sign between the two terms, we will use the ( a + b ) 2 ( a + b ) 2 pattern.

Notice that the first term of x 2 + 6 x x 2 + 6 x is a square, x 2 x 2 .

We now know a = x a = x .

What number can we add to x 2 + 6 x x 2 + 6 x to make a perfect square trinomial?

The middle term of the Binomial Squares Pattern, 2 a b 2 a b , is twice the product of the two terms of the binomial. This means twice the product of x x and some number is 6 x 6 x . So, two times some number must be six. The number we need is 1 2 · 6 = 3 . 1 2 · 6 = 3 . The second term in the binomial, b , b , must be 3.

We now know b = 3 b = 3 .

Now, we just square the second term of the binomial to get the last term of the perfect square trinomial, so we square three to get the last term, nine.

We can now factor to

So, we found that adding nine to x 2 + 6 x x 2 + 6 x ‘completes the square,’ and we write it as ( x + 3 ) 2 ( x + 3 ) 2 .

Complete a square.

To complete the square of x 2 + b x x 2 + b x :

• Step 1. Identify b b , the coefficient of x x .
• Step 2. Find ( 1 2 b ) 2 ( 1 2 b ) 2 , the number to complete the square.
• Step 3. Add the ( 1 2 b ) 2 ( 1 2 b ) 2 to x 2 + b x x 2 + b x .

Example 10.14

Complete the square to make a perfect square trinomial. Then, write the result as a binomial square.

x 2 + 14 x x 2 + 14 x

Try It 10.27

Complete the square to make a perfect square trinomial. Write the result as a binomial square.

y 2 + 12 y y 2 + 12 y

Try It 10.28

z 2 + 8 z z 2 + 8 z

Example 10.15

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared. m 2 − 26 m m 2 − 26 m

Try It 10.29

a 2 − 20 a a 2 − 20 a

Try It 10.30

b 2 − 4 b b 2 − 4 b

Example 10.16

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.

u 2 − 9 u u 2 − 9 u

Try It 10.31

m 2 − 5 m m 2 − 5 m

Try It 10.32

n 2 + 13 n n 2 + 13 n

Example 10.17

p 2 + 1 2 p p 2 + 1 2 p

Try It 10.33

p 2 + 1 4 p p 2 + 1 4 p

Try It 10.34

q 2 − 2 3 q q 2 − 2 3 q

Solve Quadratic Equations of the Form x 2 + bx + c = 0 by completing the square

In solving equations, we must always do the same thing to both sides of the equation. This is true, of course, when we solve a quadratic equation by completing the square , too. When we add a term to one side of the equation to make a perfect square trinomial, we must also add the same term to the other side of the equation.

For example, if we start with the equation x 2 + 6 x = 40 x 2 + 6 x = 40 and we want to complete the square on the left, we will add nine to both sides of the equation.

Then, we factor on the left and simplify on the right.

Now the equation is in the form to solve using the Square Root Property. Completing the square is a way to transform an equation into the form we need to be able to use the Square Root Property.

Example 10.18

How to solve a quadratic equation of the form x 2 + b x + c = 0 x 2 + b x + c = 0 by completing the square.

Solve x 2 + 8 x = 48 x 2 + 8 x = 48 by completing the square.

Try It 10.35

Solve c 2 + 4 c = 5 c 2 + 4 c = 5 by completing the square.

Try It 10.36

Solve d 2 + 10 d = −9 d 2 + 10 d = −9 by completing the square.

Solve a quadratic equation of the form x 2 + b x + c = 0 x 2 + b x + c = 0 by completing the square.

• Step 1. Isolate the variable terms on one side and the constant terms on the other.
• Step 2. Find ( 1 2 · b ) 2 ( 1 2 · b ) 2 , the number to complete the square. Add it to both sides of the equation.
• Step 3. Factor the perfect square trinomial as a binomial square.
• Step 4. Use the Square Root Property.
• Step 5. Simplify the radical and then solve the two resulting equations.
• Step 6. Check the solutions.

Example 10.19

Solve y 2 − 6 y = 16 y 2 − 6 y = 16 by completing the square.

Try It 10.37

Solve r 2 − 4 r = 12 r 2 − 4 r = 12 by completing the square.

Try It 10.38

Solve t 2 − 10 t = 11 t 2 − 10 t = 11 by completing the square.

Example 10.20

Solve x 2 + 4 x = −21 x 2 + 4 x = −21 by completing the square.

Try It 10.39

Solve y 2 − 10 y = −35 y 2 − 10 y = −35 by completing the square.

Try It 10.40

Solve z 2 + 8 z = −19 z 2 + 8 z = −19 by completing the square.

In the previous example, there was no real solution because ( x + k ) 2 ( x + k ) 2 was equal to a negative number.

Example 10.21

Solve p 2 − 18 p = −6 p 2 − 18 p = −6 by completing the square.

Another way to check this would be to use a calculator. Evaluate p 2 − 18 p p 2 − 18 p for both of the solutions. The answer should be −6 −6 .

Try It 10.41

Solve x 2 − 16 x = −16 x 2 − 16 x = −16 by completing the square.

Try It 10.42

Solve y 2 + 8 y = 11 y 2 + 8 y = 11 by completing the square.

We will start the next example by isolating the variable terms on the left side of the equation.

Example 10.22

Solve x 2 + 10 x + 4 = 15 x 2 + 10 x + 4 = 15 by completing the square.

Try It 10.43

Solve a 2 + 4 a + 9 = 30 a 2 + 4 a + 9 = 30 by completing the square.

Try It 10.44

Solve b 2 + 8 b − 4 = 16 b 2 + 8 b − 4 = 16 by completing the square.

To solve the next equation, we must first collect all the variable terms to the left side of the equation. Then, we proceed as we did in the previous examples.

Example 10.23

Solve n 2 = 3 n + 11 n 2 = 3 n + 11 by completing the square.

Try It 10.45

Solve p 2 = 5 p + 9 p 2 = 5 p + 9 by completing the square.

Try It 10.46

Solve q 2 = 7 q − 3 q 2 = 7 q − 3 by completing the square.

Notice that the left side of the next equation is in factored form. But the right side is not zero, so we cannot use the Zero Product Property. Instead, we multiply the factors and then put the equation into the standard form to solve by completing the square.

Example 10.24

Solve ( x − 3 ) ( x + 5 ) = 9 ( x − 3 ) ( x + 5 ) = 9 by completing the square.

Try It 10.47

Solve ( c − 2 ) ( c + 8 ) = 7 ( c − 2 ) ( c + 8 ) = 7 by completing the square.

Try It 10.48

Solve ( d − 7 ) ( d + 3 ) = 56 ( d − 7 ) ( d + 3 ) = 56 by completing the square.

Solve Quadratic Equations of the form ax 2 + bx + c = 0 by completing the square

The process of completing the square works best when the leading coefficient is one, so the left side of the equation is of the form x 2 + b x + c x 2 + b x + c . If the x 2 x 2 term has a coefficient, we take some preliminary steps to make the coefficient equal to one.

Sometimes the coefficient can be factored from all three terms of the trinomial. This will be our strategy in the next example.

Example 10.25

Solve 3 x 2 − 12 x − 15 = 0 3 x 2 − 12 x − 15 = 0 by completing the square.

To complete the square, we need the coefficient of x 2 x 2 to be one. If we factor out the coefficient of x 2 x 2 as a common factor, we can continue with solving the equation by completing the square.

Try It 10.49

Solve 2 m 2 + 16 m − 8 = 0 2 m 2 + 16 m − 8 = 0 by completing the square.

Try It 10.50

Solve 4 n 2 − 24 n − 56 = 8 4 n 2 − 24 n − 56 = 8 by completing the square.

To complete the square, the leading coefficient must be one. When the leading coefficient is not a factor of all the terms, we will divide both sides of the equation by the leading coefficient. This will give us a fraction for the second coefficient. We have already seen how to complete the square with fractions in this section.

Example 10.26

Solve 2 x 2 − 3 x = 20 2 x 2 − 3 x = 20 by completing the square.

Again, our first step will be to make the coefficient of x 2 x 2 be one. By dividing both sides of the equation by the coefficient of x 2 x 2 , we can then continue with solving the equation by completing the square.

Try It 10.51

Solve 3 r 2 − 2 r = 21 3 r 2 − 2 r = 21 by completing the square.

Try It 10.52

Solve 4 t 2 + 2 t = 20 4 t 2 + 2 t = 20 by completing the square.

Example 10.27

Solve 3 x 2 + 2 x = 4 3 x 2 + 2 x = 4 by completing the square.

Try It 10.53

Solve 4 x 2 + 3 x = 12 4 x 2 + 3 x = 12 by completing the square.

Try It 10.54

Solve 5 y 2 + 3 y = 10 5 y 2 + 3 y = 10 by completing the square.

Access these online resources for additional instruction and practice with solving quadratic equations by completing the square:

• Introduction to the method of completing the square
• How to Solve By Completing the Square

Section 10.2 Exercises

Practice makes perfect.

Complete the Square of a Binomial Expression

In the following exercises, complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.

a 2 + 10 a a 2 + 10 a

b 2 + 12 b b 2 + 12 b

m 2 + 18 m m 2 + 18 m

n 2 + 16 n n 2 + 16 n

m 2 − 24 m m 2 − 24 m

n 2 − 16 n n 2 − 16 n

p 2 − 22 p p 2 − 22 p

q 2 − 6 q q 2 − 6 q

x 2 − 9 x x 2 − 9 x

y 2 + 11 y y 2 + 11 y

p 2 − 1 3 p p 2 − 1 3 p

q 2 + 3 4 q q 2 + 3 4 q

Solve Quadratic Equations of the Form x 2 + b x + c = 0 x 2 + b x + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

v 2 + 6 v = 40 v 2 + 6 v = 40

w 2 + 8 w = 65 w 2 + 8 w = 65

u 2 + 2 u = 3 u 2 + 2 u = 3

z 2 + 12 z = −11 z 2 + 12 z = −11

c 2 − 12 c = 13 c 2 − 12 c = 13

d 2 − 8 d = 9 d 2 − 8 d = 9

x 2 − 20 x = 21 x 2 − 20 x = 21

y 2 − 2 y = 8 y 2 − 2 y = 8

m 2 + 4 m = −44 m 2 + 4 m = −44

n 2 − 2 n = −3 n 2 − 2 n = −3

r 2 + 6 r = −11 r 2 + 6 r = −11

t 2 − 14 t = −50 t 2 − 14 t = −50

a 2 − 10 a = −5 a 2 − 10 a = −5

b 2 + 6 b = 41 b 2 + 6 b = 41

u 2 − 14 u + 12 = −1 u 2 − 14 u + 12 = −1

z 2 + 2 z − 5 = 2 z 2 + 2 z − 5 = 2

v 2 = 9 v + 2 v 2 = 9 v + 2

w 2 = 5 w − 1 w 2 = 5 w − 1

( x + 6 ) ( x − 2 ) = 9 ( x + 6 ) ( x − 2 ) = 9

( y + 9 ) ( y + 7 ) = 79 ( y + 9 ) ( y + 7 ) = 79

Solve Quadratic Equations of the Form a x 2 + b x + c = 0 a x 2 + b x + c = 0 by Completing the Square

3 m 2 + 30 m − 27 = 6 3 m 2 + 30 m − 27 = 6

2 n 2 + 4 n − 26 = 0 2 n 2 + 4 n − 26 = 0

2 c 2 + c = 6 2 c 2 + c = 6

3 d 2 − 4 d = 15 3 d 2 − 4 d = 15

2 p 2 + 7 p = 14 2 p 2 + 7 p = 14

3 q 2 − 5 q = 9 3 q 2 − 5 q = 9

Everyday Math

Rafi is designing a rectangular playground to have an area of 320 square feet. He wants one side of the playground to be four feet longer than the other side. Solve the equation p 2 + 4 p = 320 p 2 + 4 p = 320 for p p , the length of one side of the playground. What is the length of the other side?

Yvette wants to put a square swimming pool in the corner of her backyard. She will have a 3 foot deck on the south side of the pool and a 9 foot deck on the west side of the pool. She has a total area of 1080 square feet for the pool and two decks. Solve the equation ( s + 3 ) ( s + 9 ) = 1080 ( s + 3 ) ( s + 9 ) = 1080 for s s , the length of a side of the pool.

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quadratic-equation-solve-by-completing-square-calculator

• High School Math Solutions – Quadratic Equations Calculator, Part 3 On the last post we covered completing the square (see link). It is pretty strait forward if you follow all the... Read More

Completing the Square

" Completing the Square " is where we ...

But if you have time, let me show you how to " Complete the Square " yourself.

Say we have a simple expression like x 2 + bx . Having x twice in the same expression can make life hard. What can we do?

Well, with a little inspiration from Geometry we can convert it, like this:

As you can see x 2 + bx can be rearranged nearly into a square ...

... and we can complete the square with (b/2) 2

In Algebra it looks like this:

So, by adding (b/2) 2 we can complete the square.

The result of (x+b/2) 2 has x only once , which is easier to use.

Keeping the Balance

Now ... we can't just add (b/2) 2 without also subtracting it too! Otherwise the whole value changes.

So let's see how to do it properly with an example:

The result:

x 2 + 6x + 7   =   (x+3) 2 − 2

And now x only appears once, and our job is done!

A Shortcut Approach

Here is a method you may like, it is quick when you get used to it.

First think about the result we want: (x+d) 2 + e

After expanding (x+d) 2 we get: x 2 + 2dx + d 2 + e

Now see if we can turn our example into that form to discover d and e.

Example: try to fit x 2 + 6x + 7 into x 2 + 2dx + d 2 + e

Now we can "force" an answer:

• We know that 6x must end up as 2dx , so d must be 3
• Next we see that 7 must become d 2 + e = 9 + e , so e must be −2

And we get the same result (x+3) 2 − 2 as above!

Now, let us look at a useful application: solving Quadratic Equations ...

Solving General Quadratic Equations by Completing the Square

We can complete the square to solve a Quadratic Equation (find where it is equal to zero).

But a general Quadratic Equation may have a coefficient of a in front of x 2 :

ax 2 + bx + c = 0

To deal with that we divide the whole equation by "a" first, then carry on:

x 2 + (b/a)x + c/a = 0

Now we can solve a Quadratic Equation in 5 steps:

• Step 1 Divide all terms by a (the coefficient of x 2 ).
• Step 2 Move the number term ( c/a ) to the right side of the equation.
• Step 3 Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.

We now have something that looks like (x + p) 2 = q, which can be solved this way:

• Step 4 Take the square root on both sides of the equation.
• Step 5 Subtract the number that remains on the left side of the equation to find x .

OK, some examples will help!

Example 1: Solve x 2 + 4x + 1 = 0

Step 1 can be skipped in this example since the coefficient of x 2 is 1

Step 2 Move the number term to the right side of the equation:

Step 3 Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation.

(b/2) 2 = (4/2) 2 = 2 2 = 4

Step 4 Take the square root on both sides of the equation:

Step 5 Subtract 2 from both sides:

Example 2: Solve 5x 2 – 4x – 2 = 0

Step 1 Divide all terms by 5

Step 3 Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation:

(b/2) 2 = (0.8/2) 2 = 0.4 2 = 0.16

Step 5 Subtract (-0.4) from both sides (in other words, add 0.4):

Why "Complete the Square"?

Why complete the square when we can just use the Quadratic Formula to solve a Quadratic Equation?

Well, one reason is given above, where the new form not only shows us the vertex, but makes it easier to solve.

There are also times when the form ax 2 + bx + c may be part of a larger question and rearranging it as a(x+ d ) 2 + e makes the solution easier, because x only appears once.

For example "x" may itself be a function (like cos(z) ) and rearranging it may open up a path to a better solution.

Also Completing the Square is the first step in the Derivation of the Quadratic Formula

Just think of it as another tool in your mathematics toolbox.

Footnote: Values of "d" and "e"

How did I get the values of d and e from the top of the page?

The Red Square and beyond: a guide to Moscow’s neighbourhoods

Apr 23, 2019 • 6 min read

The Red Square, Kremlin and St Basil's Cathedral in Moscow at night © Mordolff / Getty Images

One of the world’s largest cities, Moscow is a true metropolis whose ancient neighbourhoods are interspersed with newly built high-rises, inhabited by people from all over the former Soviet Union.

It’s also the city of rings: the innermost is the Kremlin itself; further away are the former defensive rings, Boulevard Ring and Garden Ring; still further are the Third Ring Road and the MKAD, which delineates the city’s borders. There’s an ongoing joke that Moscow Mayor is the Lord of the Rings. Most sights are contained within the Garden Ring, although for some more authentic neighbourhoods one has to venture further out. To help you explore Moscow’s diversity, we picked our favourite ’hoods – but this list is by no means exhaustive.

The Red Square and around

It can be argued that Moscow, or even the whole of Russia, starts at the  Red Square  – it’s an absolute must-see for any visitor. After standing in line to check out Lenin’s granite  mausoleum , go to GUM , Moscow's oldest department store. Full of luxury shops, it’s famous for the glass roof designed by one of Russia’s most celebrated architects, Vladimir Shukhov. Apart from architectural wonders, GUM has several places to eat including the Soviet-style cafeteria Stolovaya No 57 where you can sample mysterious-sounding delicacies such as the ‘herring in a fur coat’.

On the opposite side of GUM, Kremlin ’s walls and towers rise above the Red Square. Walk through the Alexander Garden  and past the grotto to the Kremlin’s entrance. It’s a treasure trove for any art and history lover: ancient gold-domed churches, icons galore and the resting place of Moscow tsars.

On the other side of the Red Square is Moscow's symbol,  St Basil's Cathedral with its multi-colored domes. Right behind it is the newly built Zaryadye Park , which showcases flora from all over Russia; another attraction is the floating bridge jutting out above the embankment and the Moscow river. A glass pavilion nearby hosts Voskhod , a space-themed restaurant with dishes from all 15 former Soviet Union republics. It’s a perfect spot for a classy evening meal and there’s often live music.

Patriarch’s Ponds

The Patriarch’s Ponds (aka Patriki) is a historical neighbourhood, celebrated in Mikhail Bulgakov’s novel Master and Margarita . Located right off Tverskaya street, Moscow’s main thoroughfare, Patriki recently became the city’s most happening quarter. It has some of the most elegant architecture, including several buildings by art-nouveau genius Fyodor Shekhtel. Narrow streets here have a cozy feel, with recently widened sidewalks and bike lanes. In the summer it becomes party central.

Start by checking out free exhibitions or one of the cutting-edge performances at the experimental theatre Praktika . But make no mistake, the neighbourhood’s main attraction are its bars and restaurants. Patriki’s residents are well-off Russians and expats, so it’s no wonder that  Moscow’s recent culinary revolution started here. Uilliam’s , one of the pioneers of this foodie movement, still rules over the scene with its floor-to-ceiling windows. Also try AQ Chicken  for everything chicken-related, Patara  for a taste of Georgian cuisine, and Cutfish for some great sushi. Finish your gastronomic tour with original cocktails at Pinch or the Moscow outpost of NYC restaurant Saxon+Parole .

Around Kursky train station

For a long time, Kursky train station was surrounded by semi-abandoned factories and the area was best avoided. It all changed in the late 2000s, when a dilapidated wine factory was turned into Winzavod , a mecca for fans of contemporary art. Today these red-brick buildings are occupied by some of Moscow’s leading galleries. After taking in all the art, pop in the small wine bar Barrell  for a glass from burgeoning wineries of Russia’s south or grab a bite at Khitrye Lyudi  cafe.

Right next to Winzavod is Artplay , another refurbished factory full of design and furniture shops and large exhibition spaces.  It’s also home to  Pluton , one of the latest additions to Moscow’s dance scene. Other Pluton residents are the multimedia art gallery Proun  and another lunch option,  Shanhaika , with authentic Chinese cuisine.

A short walk away is Arma, where a cluster of circular gas holders has been turned into offices, restaurants and clubs including Gazgolder  (it belongs to one of Russia’s most famous rappers, Basta). Apart from hip-hop concerts, Gazgolder organises regular techno parties that sometimes go non-stop from Friday to Monday.

If you’re interested in religious architecture, Taganka is the place to go. First of all, see the old Moscow at Krutitskoye Podvorye – one of those places where nothing seems to have changed in centuries. The monastery was founded in the 13th century, but in the 16th century it became the home of Moscow metropolitans and most of the surviving buildings are from that epoch. Take a tour of the grounds, and don’t miss the interior and icons of the Assumption Cathedral.

Your next stop is the Rogozhskoe settlement of ‘old believers’, a branch that split from Russian Orthodoxy in the 17th century. The settlement is dominated by an 80m-tall bell tower. The yellow-coloured Intercession Church, built in neoclassical style with baroque elements, has an important collection of icons. Next to the church grounds is the popular Trapeznaya cafeteria, with Russian food cooked using traditional recipes – a perfect spot for lunch.

A short ride away is Andronikov Monastery, which today houses the Rublyov Museum  in the old monks’ quarters. There’s a great collection of ancient Orthodox icons although none by Andrei Rublyov, who was a monk here in the 15th century. The main attraction at the monastery is the small Saviour’s Cathedral, considered the oldest surviving church in Moscow.

Finish the day at the craft-beer cluster around Taganskaya metro station. Varka offers both Russian and imported labels, with the Burger Heroes stand serving arguably the best burgers in town. Craft & Draft looks more like a respectable old-fashioned pub, with decent food, 20 beers on tap and a hundred types of bottled brews.

Khamovniki is Moscow’s ancient textile district, named after the word kham  (a type of cloth). Two main thoroughfares, Ostozhenka and Prechistenka, cut through the neighbourhood parallel to each other. The former turned into the so-called ‘Golden Mile’ of Moscow in the 1990s, with the highest real-estate prices and some of the best examples of new Russian architecture, while the latter is still mostly lined up with impressive 19th-century mansions.

Khamovniki is somewhat of a literary quarter, as several museums devoted to Russia’s best-known writers – among them  Leo Tolstoy , Alexander Pushkin  and Ivan Turgenev – popped up here during the last century. There’s also plenty to see for an art lover. The  Multimedia Art Museum regularly hosts exhibitions by some of the best photographers from all over the world, as well as contemporary art. Several galleries, including RuArts  and Kournikova Gallery , have also found home in Khamovniki.

When you’ve had your fill of literature and art, stop by Gorod Sad on Ostozhenka, an outpost of a local health-food chain, and order dishes such as pumpkin soup or grilled vegetables salads. Afterwards, head to Dom 12 , which is located just off Ostozhenka street. This restaurant and wine bar is frequented by the city’s intellectuals and its schedule includes lectures, book presentations and film screenings, while in the summer guests migrate to a lovely courtyard.

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Solve the quadratic equation by completing the square. x2 + 4x - 3 = 0 A) x = 4 ±7 B) x = 2 ±7 C) x = -4 ±7 D) x = -2 ±7

the answer is D i just did it on usa test prep

Step-by-step explanation:

📚 Related Questions

• Robin needed 3 2/3 feet of thread to finish a pillow she was making. if she has 2 times as much thread as she needz what is the length of the thread she has

Solve for x. Use the completing the square method. 4x^2+8x=7 Round decimals to the nearest tenth. Enter your answers in the boxes. x = BOX or x = BOX

[tex]\displaystyle -2,66 ≈ x\:or\:0,66 ≈ x[/tex]

Step-by-step explanation :

All work is shown above.

I am joyous to assist you anytime.

In 2010, a city population was 55,210 and it was declining at a rate of 1.09 each year. What is the best prediction for when the city population will first be below 48,000? 2019 2022 2025 2028

the correct answer is 2022  

just took the test and this is 100% correct

identify the parent function: f(x)=7x^2-5x+3 quadratic function linear function rational function exponential function

Details : identify the parent function: f(x)=7x^2-5x+3quadratic functionlinear

identify the parent function: f(x)= 1/2x- 9 4/5 linear function exponential function rational function quadratic function

• Mary buys a spool of thread for sewing. there are 7 m of thread on a then spool. she uses 210 cm. how much is left on the spool in centimeters?

1 m = 100 cm

• You can find the density of an object by using D=m/v , where image the mass of the object in grams and V is the volume in millimeters. What is the density of the can of tomato sauce with a diameter of 10 cm, height 11.5 cm, in a mass of 822 grams? Use the conversion factor of 1 cm^3= 1mL. Please helppp

Kim has joined a music club. she pays a fee of $5.00 each month and then can download songs for $0.58 each. if she can spend no more than $13.00 this month, how many songs can she download?

Details : Kim has joined a music club. she pays a fee of $5.00 each month and

Joelle earns her regular pay of 7.50 per hour for up to 40 hours of work in a week. for each hour over 49 hours of work in a week, joelle is paid 1.5 times her regular pay. how much does joelle earn for a week in which she works 42 hours

Jennifer has a bag of chips that contains 2 red chips and 1 black chip. if she also has a fair die, what is the probability that she rolls an odd number and pulls a red chip?

Dividing one number by another results in the _____ of one number to the other.

Yvette uses 6 grams of tea leaves to make 24 fluid ounces of tea. last week she made 288 fluid ounces of tea. how many grams of tea leaves did yvette use to make tea last week?

Details : Yvette uses 6 grams of tea leaves to make 24 fluid ounces of tea.

Virginia purchased 3 packets of flower seeds to plant in her garden. if forty-five flowers bloomed, and each packet contained the same amount of seeds, what is the constant of proportionality?

Suppose 16% of the listeners of a radio station listen to it while they are at work. What is the approximate standard deviation of the sampling distribution of the proportion for the sample of size 72

Answer with explanation:

Sample Size (n)=72

→P=% of people who listen to radio while they are at work =16 %

→B=1 - P= 100% - 16% =84%

→Standard Deviation when Probability and Sample Size is given

  [tex]\sigma=\sqrt{\frac{P(1-P)}{n}}\\\\ \sigma =\sqrt{\frac{\frac{16}{100} \times \frac{84}{100}}{72}}\\\\ \sigma=\sqrt{\frac{1344}{720,000}}\\\\\sigma=\sqrt{\frac{18.66}{10000}}\\\\\sigma=\frac{4.320}{100}\\\\ \sigma=0.0432[/tex]

So, standard deviation of the sampling distribution of the proportion for the sample of size 72= 0.0432=0.043 (Approx)

so it's 4.3%?

x^2 + 5x - 36 Which binomial is a factor of the polynomial? A) (x + 6) B) (x + 9) C) (x + 5) D) (x + 12)

To enter a zoo, adults must pay $5. whereas, children and seniors pay half price. on one day, the zoo collected a total of $765. if the zoo had 223 visitors that day, how many half-price admissions and how many full price admissions did the zoo collect

Details : To enter a zoo, adults must pay $5. whereas, children and seniors

Write the ordered pair that represents yz. Then find the magnitude of yz . y=14,-23 z=23,-14

Ordered pair of yz is ( 9 , 9 ) and Magnitude of yz is 9√2 unit.

Given ordered pair of y and z are ( 14 , -23 ) and ( 23 , -14 ) respectively

To find: Ordered pair of yz and magnitude of yz

To find order pair of yz we subtract ordered pair of y from z.

Ordered pair of yz = ( 23 - 14 , -14 - (-23) )

                               = ( 9 , -14 + 23 )

                               = ( 9 , 9 )

Magnitude of yz , [tex][\left|yz\right|\:=\:\sqrt{9^2+9^2}[/tex]

                                    = [tex]\sqrt{81+81}[/tex]

                                    = [tex]\sqrt{2\times81}[/tex]

                                    = [tex]9\sqrt{2}[/tex]

Therefore, Ordered pair of yz is ( 9 , 9 ) and Magnitude of yz is 9√2 unit.

This operation is used for the following terms: less than, minus, difference, reduced, decreased by, diminished by.

Subtraction.

We are asked to determine the name of operation that is used for given terms: less than, minus, difference, reduced, decreased by, diminished by.

We can see that all the given terms show that some quantity is being taken away from them. We know that when we take some quantity from another quantity, this means we are using subtraction.

Therefore, the name of the operation that defines the given terms is "subtraction".

There are 25 students who answered a survey about sports. four-fifths of them like football. how many students like football?

20 students like football

We know that:

• The total students are 25.
• Four-fifth from the total like football.

So, to solve this problem we just have to calculate the four-fifth of 25. Four-fifth refers to fraction:

[tex]\frac{4}{5} 25=4(5)=20[/tex]

Therefore, 20 students like football, according to the survey about sport.

A plumber charges a customer 60 dollars an hour plus the price of any parts needed to complete the work. If the total bill is 240 dollars and just the parts alone cost 90 dollars, how long did it take to complete the work. 4 hours 300 minutes 2.5 hours 6.17 hours

Details : A plumber charges a customer 60 dollars an hour plus the price of

If a line falls on the points (7, 3) and (4, 2), what is its slope? enter your answer as a fraction in lowest terms. use a slash mark ( / ) as the fraction bar.

The lengths of three sides of a triangle are m units, n units, and p units, respectively. which inequality must be true?

B) n < m + p

The question is incomplete. Here are the options for this question :

A) m > n + p

C) p > m + n

D) p < m - n

For a triangle of sides a, b and c the sum of two sides must be greater to the remaining side. Therefore, we have the following equations :

a + b > c

a + c > b

b + c > a

For the exercise, the first option A) is incorrect. It should be m < n + p

Otherwise, the option B) is correct.

The option C) has the same problem that option A).

It should be p < m + n

Option D) p < m - n can be rearrange as : p + n < m that is wrong. It should be p + n > m

The only correct option is B) n < m + p

Henry gargles with mouthwash. which is a reasonable amount of mouthwash for henry to use. a. 2 fl oz. b.8 fl oz. c. 12 fl oz. d 20 fl oz.

One aquarium is 12 inches long,another is 15 inches long, and a third is 18 inches long.they are all 18 inches deep and 12inches wide .which aquarium can hold exactly 3240 cubic inches of water

Details : One aquarium is 12 inches long,another is 15 inches long, and a third

On saturday rhonda filled 3 1/3 buckets with shells . on sunday she filled 4 2/3 buckets with shells .how many more buckets of shells. rhonda fill on sunday ? write the answer.in simplest form .

One number is 1/2 of another number. The sum of the two numbers is 21. Find the two numbers. Use a comma to separate your answers

The numbers are 7, 14.

What is a Ratio?

A ratio indicates the number of times one number contains another.

Let the first number be x.

Given that one number is 1/2 of another number , therefore, the second number is:

The sum of the number is 21, therefore,

x + x(1/2) = 21

(3/2)x = 21

x = (21 ×2)/3

The other number is 1/2(14) = 7.

Hence, the numbers are 7, 14.

To learn more about ratios , click here:

https://brainly.com/question/13419413

Grace took five bags of aluminum cans to the recycling center. each bag held an average of 300 cans. of the recycling center pays $2.00 for every 450 cans, about how much money will grace get?

I literally JUST took a test that had that question. I got it right!!

Nigel borrows $250 and pays 5.5% simple interest each year. if he pays back the money in one year, what is the total amount that he pay?

Details : Nigel borrows $250 and pays 5.5% simple interest each year. if he

Leorys favorite basketball player is 6 feet 8 inches tall how tall i leorys favorite basketball player in inches

Katia bought two melons. she says the difference in mass between a melon is 5000 g. which two melons did katia buy?watermelons 8 kg cantaloupe 5 kg honeydew 3 kg ciba melon 2 kg or crenshaw melon 1 kg

She bought a watermelon and a honeydew.

As katia bought two different melons, with a difference in mass between the two of 5000g, this converted into kilograms would be 5kg.

So you´d have an equation like this

Melon1 - Melon 2= 5kg

So you have 5 melons you just try them on the equation

Watermelon 8 - Crenshaw melon 1=5

Since 7 isn´t the same as 5 it is incorrect.

So your answer would be

Watermelon 8- Honeydew 3= 5

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Drone strikes Moscow building as region hit by sixth successive night of attacks

Building under construction in capital’s financial district damaged as three people killed in Belgorod region, governor says

• See all our Ukraine war coverage

A drone hit a building under construction in Moscow’s financial district early on Wednesday in the sixth straight night of aerial attacks on Russia’s capital region.

A loud explosion was heard in Moscow’s business district on Wednesday morning, a short time after flights were suspended at the city’s airports, Russia’s RIA news agency reported. Russian media published videos showing the moment of the explosion that left charred holes in the side of the buildings.

The central district is less than three miles from the Kremlin.

The mayor of Moscow, Sergei Sobyanin, said the Russian military downed two more drones over the western part of the Moscow region.

Ukraine has launched a campaign of drone strikes targeting the Russian capital in recent days as Kyiv seeks to demonstrate its ability to hit Moscow and to keep the Kremlin’s war in the hearts and minds of the Russian elites and others seeking to ignore the invasion of Ukraine .

Earlier waves of drone strikes have hit the Moscow city financial centre, other residential buildings, or targeted areas in the wealthy western suburbs of the Rublyovka district , a few miles from Vladimir Putin’s Novo-Ogaryovo residence. Though the drone attacks on Moscow have occurred almost daily in recent weeks, they have caused little damage and no casualties.

They have, however, triggered travel chaos, with big airports around the Russian capital forced to repeatedly close for departing and arriving flights.

Also on Wednesday, the governor of the Belgorod region, which borders Ukraine , said a Ukrainian drone had hit a sanatorium in a village. He said two people had died on the spot and doctors had been unable to save a third. If confirmed, the drone attack would mark the first known incident involving civilian deaths on Russian territory.

Ukraine typically declines to claim responsibility for attacks on Russian territory, although Kyiv officials have frequently celebrated such attacks with cryptic or mocking remarks. Last month, Ukraine’s president, Volodymyr Zelenskiy, said that attacks on Russian territory were an “inevitable, natural and absolutely fair process” of the war between the two countries.

In response to reports of recent drone attacks, the US state department said Washington did not encourage or enable the strikes on Russian territory. “It is up to Ukraine to decide how it chooses to defend itself from the Russian invasion,” the US spokesperson said.

On Tuesday, British military intelligence said a Ukrainian drone attack appeared to have destroyed a supersonic Russian bomber on the Soltsy-2 airbase, 400 miles (650km) from the border with Ukraine.

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Social media images showed an aircraft that resembled the Tu-22M3 in flames on a runway. The planes have been used regularly in campaigns that killed civilians in Ukraine.

Ukraine on Wednesday also said it had destroyed a Russian S-400 anti-aircraft system on the Russian-occupied Crimean peninsula.

“At about 10am (0700 GMT) an explosion occurred … destroying a Russian long- and medium-range S-400 Triumph air defence system,” the Ukrainian defence ministry said on social media.

The ministry published a video of a large explosion with a huge column of smoke billowing into the sky. “This is a painful blow to the occupiers’ air defence system,” the ministry said.

Russia’s defence ministry did not comment but several pro-Moscow bloggers said the destruction of the anti-aircraft system exposed vulnerabilities in Russian defences.

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