Completing the Square

Section 9.1MATH 116-460

Mr. Keltner

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

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.

Completing the Squarein 3 “Easy” Steps

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.

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?

Solving Quadratic Equations by Completing the SquareRelated 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.

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 + 150 = 0 by completing the square.

Example 5• Solve the equation by

completing the square. (NOTE: You cannot apply the Zero-Product

Property and say that either 6x = 12 or (x + 8) = 12 and solve.)

6x (x + 8) = 12

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 + 3Find 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.

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 + 3Find 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.

Fountain Example• At the Buckingham

Fountain in Chicago, the water’s height h (in feet) above the main nozzle can be modeled by h = -16t2 + 89.6t, 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?

AssessmentPgs. 628-631:

#’s 7-98, multiples of 7