1. Quadratic Formula

2. Solving Quadratics Completing the Square

3. 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 + 9
x2 - 10x + 25
x2 + 12x + 36
Try to factor these (they’re easy).

4. Perfect Square Trinomials
x2 + 6x + 9
x2 - 10x + 25
x2 + 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

5. 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

6. 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

7. Perfect Square Trinomials
Create perfect square trinomials.
x2 + 20x + ___
x2 - 4x + ___
x2 + 5x + ___
100 4
25/4

8. Completing the Square a) b) c) d) Example 2:
What number should be added to Complete The Square?
Now how do you write each as a perfect square? a) b) c) d)

9. Methods for Solving Quadratic Equations
10.5 Completing the Square
Methods for Solving Quadratic Equations
Factor
Square Root
Completing the Square
Quadratic Formula

10. Square Root Property

11. Completing the Square
Solve: = =

12. Completing the Square
Solve:

13. Solving Quadratic Equations by Completing the Square
Try the following examples. Do your work on your paper
and then check your answers.

14. 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.

15. Standard to Graphing: Quadratic
Find the vertex of the following equation by completing the square:
y = x2 + 8x + 25
Find the “c” that completes the square
GOAL
y = a ( x – h ) k
Complete the Square:
Add to both sides
y = (x2 + 8x ) + 25 16 16
Factor what is in the Parentheses
y = (x + 4)2 16
+ 25
y = (x + 4) 2 - 9
Simplify
(-4, 9)
Vertex:

16. Standard to Graphing: Quadratic
Find the vertex of the following equation by completing the square:
y = 3x2 – 18x – 10
GOAL
y = a ( x – h ) k
y = 3(x2 – 6x ) - 10
+ 3 9 9
y = (x – 3)2
– 10 – 27
y = 3(x – 3)2 – 37
(3,-37)
Vertex:

17. ( ) 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.
Add
–10 2 ( ) =
(–5) 25
to each side.
y + 25 = (x2 – 10x + 25) + 22
y + 25 = (x – 5)2 + 22
Write x2 – 10x + 25 as a binomial squared.
y = (x – 5)2 - 3
Write in vertex form.
The vertex form of the function is y = (x – 5)2 - 3
The vertex is (5, –3).
ANSWER

18. GUIDED PRACTICE
for Examples 6 and 7
Write the quadratic function in vertex form. Then identify the vertex.
13.
y = x2 – 8x + 17
y - 1 = (x – 4)2 ; (4, 1).
ANSWER
14.
y = x2 + 6x + 3
y + 6 = (x + 3)2 ; (–3, –6)
ANSWER
15.
f(x) = x2 – 4x – 4
ANSWER
y + 8 = (x – 2)2 ; (2 , –8)

19. EXAMPLE 7
Find the maximum value of a quadratic function
Baseball
The height y (in feet) of a baseball t seconds after it is hit is given by this function:
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.

20. Find the maximum value of a quadratic function
EXAMPLE 7
Find the maximum value of a quadratic function
y = –16t2 + 96t +3
Write original function.
y = –16(t2 – 6t) +3
Factor –16 from first two terms.
y + ? = –16(t2 –6t + ? ) + 3
Prepare to complete the square.
y +(–16)(9) = –16(t2 – 6t + 9 ) + 3
Add
to each side.
(–16)(9)
y – 144 = –16(t – 3)2 + 3
Write t2 – 6t + 9 as a binomial squared.
y = –16(t – 3)
Vertex Form
The vertex is (3, 147), so the maximum height of the baseball is 147 feet.
ANSWER

21. g(x)+ = 5(x2 – 10x + ) + 128 Check It Out! Example 4b
Write the function in vertex form, and identify its vertex
g(x) = 5x2 – 50x + 128
Factor so the coefficient of x2 is 1.
g(x) = 5(x2 – 10x) + 128
Set up to complete the square.
g(x)+ = 5(x2 – 10x + ) + 128
Add
Because
is multiplied by 5,
you must add
g(x) = 5(x2 – 10x ) + 128

22. Check It Out! Example 4b Continued
g(x) = 5(x – 5)2 + 3
Simplify and factor.
Because h = 5 and k = 3, the vertex is (5, 3).
Check A graph of the function on a graphing calculator supports your answer.