Completing the Square

CCSS: A.SSE.3; F.IF.7

Solving Quadratics By

Completing the SquareMust be a perfectSquare

CCSS: A.SSE.3

CHOOSE and PRODUCE an equivalent form of an expression to REVEAL and EXPLAIN properties of the quantity represented by the expression.

a. FACTOR a quadratic expression to reveal the zeros of the function it defines.

b. COMPLETE THE SQUARE iin a quadratic expression to REVEAL the maximum or minimum value of the function it defines.

CCSS: F.IF.7

GRAPH functions expressed symbolically and SHOW key features of the graph, by hand in simple cases and using technology for more complicated cases.*

a. GRAPH linear and quadratic functions and show intercepts, maxima, and minima.

Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the

reasoning of others.   4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

Essential Questions

How do I determine the domain, range, maximum, minimum, roots, and y-intercept of a quadratic function from its graph?

How do I use quadratic functions to model data?  How do I solve a quadratic equation with non - real

roots?

2( 5) 64x 5 8x

When you take the square root, You MUST consider the Positive and Negative answers.

5 8x 5 8x 5 5

13x 5 5

3x

PerfectSquare

On One side

Take Square Root

ofBOTH SIDES

2 ( 5) 64x

PerfectSquare

On One side

Take Square Root

ofBOTH SIDES

But what happens if you DON’T have a perfect square on one side…….

You make it a Perfect Square

Use the relations on next slide…

2( 6)x ( 2 ) To expand a perfect square binomial:

2 12 36x x 6x 26

We can use these relations to find the missing term….To make it a perfect square trinomial that can be factored into a perfect square binomial.

2 _ _12 _x x 12 2 6 626 36

36

2x

Take ½ middle term

Then square it

The resulting trinomial is called a perfect square trinomial,

which can be factored into a perfect square binomial.

2 _ _18 _ _x x

18 2 92(9) 81

81 2( 9)x

1. 2 12 0x x

1. Make one side a perfect square

2. Add a blank to both sides

3. Divide “b” by 2

4. Square that answer.

5. Add it to both sides

6. Factor 1st side

7. Square root both sides

8. Solve for x

2 0x x ___ ___12 2 6

2(6) 36

36 362( 6)x 362( 6) 36x 6 6x

6 6x 6 6x 6 6

12x 6 6

0x

12

Factor this Perfect square trinomial

2 12 36x x

What is the

Square root

of x2

2( )x

Bring dow

n sign

6W

hat is the S

quare root of 36

2( 6)x

2. 2 8 0x x

1. Move constant to other side.

2. Add a blank to both sides

3. Divide “b” by 2

4. Square that answer.

5. Add it to both sides

6. Factor 1st side

7. Square root both sides

8. Solve for x

2 8x x ___ ___6 2 3

2(3) 9

9 92( 3)x 12( 3) 1x

3 1x 3 1x 3 1x 3 3

4x 3 3

2x

6

6

Factor this Perfect square trinomial

2 6 9x x

What is the

Square root

of x 2

2( )x

Bring dow

n sign

3W

hat is the S

quare root of 9

2( 3)x

3. 2 8 84 0x x

1. Move constant to other side.

2. Add a blank to both sides

3. Divide “b” by 2

4. Square that answer.

5. Add it to both sides

6. Factor 1st side

7. Square root both sides

8. Solve for x

2 84x x ___ ___8 2 4

2(4) 1616 16

2( 4)x 1002( 4) 100x

4 10x 4 10x 4 10x 4 4

14x 4 4

6x

8

Factor this Perfect square trinomial

2 8 16x x

What is the

Square root

of x 2

2( )x

Bring dow

n sign

4W

hat is the S

quare root of 9

2( 4)x

4. 2 2 15 0x x

1. Move constant to other side.

2. Add a blank to both sides

3. Divide “b” by 2

4. Square that answer.

5. Add it to both sides

6. Factor 1st side

7. Square root both sides

8. Solve for x

2 15x x ___ ___2 2 1

2(1) 11 1

2( 1)x 162( 1) 16x

1 4x 1 4x 1 4x 1 1

3x 1 1

5x

2

Factor this Perfect square trinomial

2 2 1x x

What is the

Square root

of x 2

2( )x

Bring dow

n sign

1W

hat is the S

quare root of 9

2( 1)x

Steps to solve Quadratics by completing the square:

Move the constant to side by itself. Make the side (w/variables) a perfect square by

adding a certain number to both sides. To calculate this number

– Divide “b” (middle term) by 2– Then square that answer

Take the square root of both sides of eq Then solve for x

In a perfect square, there is a relationship between the coefficient of the middle term and the constant term.

2( 7)x

7 1

(14)2

27 49

2 14 49x x

7

Creating a Perfect Square Trinomial

In the following perfect square trinomial, the constant term is missing. X2 + 14x + ____

Find the constant term by squaring half the coefficient of the linear term.

(14/2)2 X2 + 14x + 49

Perfect Square Trinomials

Create perfect square trinomials.

x2 + 20x + ___ x2 - 4x + ___ x2 + 5x + ___

100

4

25/4

Solving Quadratic Equations by Completing the Square

2

2

2

2

2

1. 2 63 0

2. 8 84 0

3. 5 24 0

4. 7 13 0

5. 3 5 6 0

x x

x x

x x

x x

x x

Try the following examples. Do your work on your paper and then check your answers.

1. 9,7

2.(6, 14)

3. 3,8

7 34.

2

5 475.

6

i

i