Quadratic Formula

Solving Quadratics

Completing the Square

The Aim of Completing the Square

… is to write a quadratic function as a perfect square.

Here are some examples of perfect squares!

x2 + 6x + 9x2 - 10x + 25x2 + 12x + 36

Try to factor these (they’re easy).

Perfect Square Trinomialsx2 + 6x + 9x2 - 10x + 25x2 + 12x + 36

Can you see a numerical connection between …

6 and 9 using 3

-10 and 25 using -5

12 and 36 using 6

=(x+3)2

=(x-5)2

=(x+6)2

The Perfect Square Connection

For a perfect square, the following relationships will always be true …

x2 + 6x + 9

x2 - 10x + 25

Half of these values squared … are these values

The Perfect Square Connection

In the following perfect square trinomial, the constant term is missing. Can you predict what it might be?

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

What number should be added to Complete The SquareComplete The Square?

a)

b)

c)

2 4x x 2( 2)x 2( 9)m

2( 6)y

4

Now how do you write each as a perfect squareperfect square?

d) 2 16a a 2( 8)a 64

Example 2:

Methods for Solving Quadratic Equations

10.5 Completing the Square

1. Factor2. Square Root3. Completing the Square4. Quadratic Formula

Square Root Property

Completing the Square

Solve: ==

Completing the Square

Solve:

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

Standard 9 Write a quadratic function in vertex form

Vertex form- Is a way of writing a quadratic equation that facilitates finding the vertex.

y – k = a(x – h)2

The h and the k represent the coordinates of the vertex in the form V(h, k).The “a” if it is positive it will mean that our parabola opens upward and if negative it will open downward.A small value for a will mean that our parabola is wider and vice versa.

y + = (x2 + 8x + ) + 25

Standard to Graphing: Quadratic

y = x2 + 8x + 25

21

82

24

16

1616

y + = (x + 4)2

Vertex: (-4, 9)

+ 25

Find the vertex of the following equation by completing the square:

y = a ( x – h )2 + k

GOAL

Complete the Square:

Find the “c” that completes the

square

Factor what is in the Parentheses

Add to both sides

Simplify

16

y = (x + 4) 2 - 9

Standard to Graphing: Quadratic

y = 3x2 – 18x – 10

y = 3(x2 – 6x + ) - 10

21

62

23

9

99

y = (x – 3)2

Vertex: (3,-37)

3 – 10 – 27

y = 3(x – 3)2 – 37

+ 3

Find the vertex of the following equation by completing the square:

y = a ( x – h )2 + kGOAL

Standard 9 Write a quadratic function in vertex form

Write y = x2 – 10x + 22 in vertex form. Then identify the vertex.

y = x2 – 10x + 22 Write original function.

y + ? = (x2 –10x + ? ) + 22 Prepare to complete the square.

y + 25 = (x2 – 10x + 25) + 22Add –102

2( ) = (–5)2= 25 to each side.

y + 25 = (x – 5)2 + 22 Write x2 – 10x + 25 as a binomial squared.

y = (x – 5)2 - 3Write in vertex form.

The vertex form of the function is y = (x – 5)2 - 3 The vertex is (5, –3).

ANSWER

GUIDED PRACTICE for Examples 6 and 7

y = x2 – 8x + 17

y - 1 = (x – 4)2 ; (4, 1).ANSWER

13.

Write the quadratic function in vertex form. Then identify the vertex.

y = x2 + 6x + 3

y + 6 = (x + 3)2 ; (–3, –6)

ANSWER

14.

f(x) = x2 – 4x – 4

y + 8 = (x – 2)2 ; (2 , –8)ANSWER

15.

EXAMPLE 7 Find the maximum value of a quadratic function

The height y (in feet) of a baseball t seconds after it is hit is given by this function:

Baseball

y = –16t2 + 96t + 3

Find the maximum height of the baseball.

SOLUTION

The maximum height of the baseball is the y-coordinate of the vertex of the parabola with the given equation.

EXAMPLE 7 Find the maximum value of a quadratic function

y = –16t2 + 96t +3 Write original function.

y + ? = –16(t2 –6t + ? ) + 3 Prepare to complete the square.

y – 144 = –16(t – 3)2 + 3 Write t2 – 6t + 9 as a binomial squared.

y = –16(t – 3)2 + 147Vertex Form

y = –16(t2 – 6t) +3 Factor –16 from first two terms.

y +(–16)(9) = –16(t2 – 6t + 9 ) + 3 Add to each side.(–16)(9)

The vertex is (3, 147), so the maximum height of the baseball is 147 feet.

ANSWER

Write the function in vertex form, and identify its vertex

Factor so the coefficient of x2 is 1.

g(x) = 5(x2 – 10x) + 128Set up to complete the square.

Add .

Because

is multiplied by 5,

you must add 5 .

Check It Out! Example 4b

g(x) = 5x2 – 50x + 128

g(x)+ = 5(x2 – 10x + ) + 128

g(x)+ = 5(x2 – 10x + ) + 128

Simplify and factor.g(x) = 5(x – 5)2 + 3

Because h = 5 and k = 3, the vertex is (5, 3).

Check It Out! Example 4b Continued

Check A graph of the function on a graphing calculator supports your answer.