Ch 10: Polynomials G) Completing the Square Objective: To solve quadratic equations by completing the square.

* * Methods for Solving Quadratic Equations 1) Square Root Method (Works when no “x” term) x = ±3 2) Graphic Method (Works when on lattice point) x=-1 x=3 * 3) Quadratic Formula (Always works) a=1 b= -2 c= -3 4) Factoring Method (Works when factorable) x = −1, x = 3 * 5) Completing the Square (Always works)

√ √ Rules from Standard Form: ax2 + bx + c = 0 Subtract c from both sides….. Divide both sides by a……… Divide the x term by 2............ Add b 2 to both sides…….. (the left side is a “perfect square”) 5) Square root both sides……….. 6) Solve for x …………………... ax2 + bx = −c ax2 + b x = −c a a a x2 + b x = −c 2 a a x2 + b x = −c + b 2 + b 2 a 2a 2a 2a 2a 2 x + b = −c + b 2 2a a 2a 2 √ x + b √ = −c + b 2 2a a 2a | x + b/(2a)| =

What number (c) makes this a Example 1 What number (c) makes this a Perfect Square? + 9 + 9 ( ) 6 2 2 = 9 (x )2 = 7 + 3 + 9

Example 2 Complete the perfect square trinomial ( ) -8 2 2 = 16 (x )2 = -5 - 4 + 16

Example 3 Complete the square

Example 4 Complete the square

y2 − 14y + c x2 + 12x + c 2 2 c = 49 c = 36 r2 + 26r + c 2t2 – 7t + c Classwork Find the value of c that completes the square 2 2 1) y2 − 14y + c 2) x2 + 12x + c 2 2 c = 49 c = 36 2 2 3) r2 + 26r + c 4) 2t2 – 7t + c 2 2 2 2 c = 49 c = 169 16

x2 – 4x – 34 = -2 x2 − 12x – 60 = 4 {-4, 8} {-4, 16} n2 + 8n – 26 = 7 Solve each equation by completing the square 5) x2 – 4x – 34 = -2 6) x2 − 12x – 60 = 4 {-4, 8} {-4, 16} 7) n2 + 8n – 26 = 7 8) 2p2 – 20p + 16 = -2 {-11, 3} {1, 9}