1. Section 9.1 MATH 116-460 Mr. Keltner
Completing the Square
Section 9.1
MATH
Mr. Keltner

2. Example 1
Solve (x + 6)2 = 20.
We know how to solve equations when it is possible to take the square root of each side, such as:
4x2 = 16 Constant multiple with a variable
x2 + 14x + 49 = 25 A quantity that can be rewritten
x2 = -76 Where we take the square root of a negative number, or use i

3. A need for a new strategy
Sometimes, we will have an expression of the form x2 + bx, that requires us to insert an additional term to make it a perfect square, like (x + k)2.
The process of forcing a quadratic expression to become a perfect square trinomial is called completing the square.

4. Completing the Square in 3 “Easy” Steps
These steps may only be used on a quadratic expression in the form ax2 + bx, where a = 1, and b is a real number.
Find half the coefficient of x.
Square that value.
Replace c with the resulting value.
Example:
Find the value of c that makes x2 -26x + c a perfect square trinomial. Then write the expression as the square of a trinomial.
-26 ÷ 2 = -13
(-13)2 = 169
x2 -26x + c becomes x2 -26x + 169, which is equal to (x-13)2 when it is factored.

5. Example 2
Find the value of c that makes each expression a perfect square trinomial. Then write each expression as the square of a binomial.
x2 + 14x + c
x2 + 22x + c
x2 – 9x + c
What do you notice about the value of c if b happens to be an odd number?
Does the sign of c change or does it remain constant?

6. Solving Quadratic Equations by Completing the Square
Related example:
Solve x2 - 4 = 12.
Your first step would be to add 4 to each side of the equation.
Just the same as we add the same value to both sides of this equation, we apply this same idea when completing the square.
Example 3
Solve x2 - 10x + 1 = 0 by completing the square.

7. Solving ax2 + bx + c = 0 if a ≠ 1
Divide every term of both sides by the coefficient of x2 (the value of a).
Make sure to balance the equation by adding the same value to both sides.
Example 4: Solve 3x2 – 36x = 0 by completing the square.

8. Example 5 Solve the equation by completing the square. 6x (x + 8) = 12
(NOTE: You cannot apply the Zero-Product Property and say that either 6x = 12 or (x + 8) = 12 and solve.)
6x (x + 8) = 12

9. Baseball example: Finding a maximum value
The height, y (in feet), of a baseball x seconds after it is hit is given by the equation:
y = -16x2 + 96x + 3
Find the maximum height of the baseball.
The maximum height of the baseball will be the y-coordinate of the vertex of the parabola.
It will help if we can write the equation in vertex form.

10. Baseball Example, Cont.
The height, y (in feet), of a baseball x seconds after it is hit is given by the equation:
y = -16x2 + 96x + 3
Find the maximum height of the baseball.
Start by writing the function in vertex form.
If we can find the y-value at the vertex, we will have found the maximum height of the ball.

11. Fountain Example
At the Buckingham Fountain in Chicago, the water’s height h (in feet) above the main nozzle can be modeled by h = -16t t, where t is the time in seconds) since the water has left the nozzle.
Find the highest point the water reaches above the fountain.
What does this vertex represent, in real-world terms?

12. Pgs. 628-631: #’s 7-98, multiples of 7
Assessment
Pgs :
#’s 7-98, multiples of 7