Solving Quadratic equations by Completing the Square 9-4 Notes for Algebra 1 Solving Quadratic equations by Completing the Square

9-4 pg. 577 10-30, 37-42, 77-81o

Completing the Square Steps for completing the square of any quadratic expression. 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 1.) Make sure the leading coefficient (𝑎) is one. 2.) Get the 𝑎𝑥 2 and 𝑏𝑥 on one side and the 𝑐 term on a side by itself. 3.) Add 𝑏 2 2 to both sides of the equation (Then factor and simplify) 4.) Solve for 𝑥.

Example 1: Complete the Square 1.) Find the value of 𝑐 that makes 𝑥 2 −12𝑥+𝑐 a perfect square trinomial.

Example 1: Complete the Square 1.) Find the value of 𝑐 that makes 𝑥 2 −12𝑥+𝑐 a perfect square trinomial. 36

Example 2: Solve an Equation by completing the square 𝑥 2 +6𝑥+5=12

Example 2: Solve an Equation by completing the square 𝑥 2 +6𝑥+5=12 −7, 1

Example 3: Equation with 𝑎≠1 −2𝑥 2 +36𝑥−10=24

Example 3: Equation with 𝑎≠1 −2𝑥 2 +36𝑥−10=24 17, 1

Example 4: Use a Graph of a Quadratic Function CANOEING Suppose the rate of flow of an 80-foot-wide river is given by the equation 𝑟=−0.01 𝑥 2 +0.8𝑥, where 𝑟 is the rate in miles per hour and 𝑥 is the distance from the shore in feet. Joel does not want to paddle his canoe against a current that is faster than 5 miles per hour. At what distance from the river bank must he paddle in order to avoid a current of 5 miles per hour?

Example 4: Use a Graph of a Quadratic Function CANOEING Suppose the rate of flow of an 80-foot-wide river is given by the equation 𝑟=−0.01 𝑥 2 +0.8𝑥, where 𝑟 is the rate in miles per hour and 𝑥 is the distance from the shore in feet. Joel does not want to paddle his canoe against a current that is faster than 5 miles per hour. At what distance from the river bank must he paddle in order to avoid a current of 5 miles per hour? Up to 7 ft. from either bank NOTE: Then solutions of the equation are about 7 ft. and about 73 ft. Since the river is 80 ft. wide, 80−73=7. Both ranges are within 7 ft. of one bank or the other.