Solving Quadratic Equations by Completing the Square Day 1: Section 9 – 3

Main Ideas Solve quadratic equations by finding the square root. Solve quadratic equations by completing the square.

Determine if the quadratic is a perfect square. x2 + 4x + 4 = x2 + 2(2)x + 22 = (x+2)2

Complete the Square 1.) m2 + 12m + 36 = m2 + 2(6)m + 62 = (m + 6)2 2.) r2 – 6r + 9 = r2 + 2(-3)r + (-3)2 = (r – 3)2 3.) x2 – 2x + 1 = x2 + 2(-1)x + (-1)2 = (x – 1)2

Solve x2 – 2x + 1 = 9 x2 – 2x + 1 = 9 x2 + 2(-1)x + (-1)2 = 9 (x – 1)2 = 9 x = 3 + 1 or x = -3 + 1 x = 4 or x = -2 Solution Set: {-2, 4}

Solve x2 – 4x + 4 = 5 x2 – 4x + 4 = 5 x2 + 2(-2)x + (-2)2 = 5 (x – 2)2 = 5 Solution Set: {-0.2, 4.2}

x = {-5, 1} x = -2 + 3 x = 1 x = -2 – 3 x = -5 1.) x2 + 4x + 4 = 9 x2 + 4x + 4 = 9 x2 + 2(x)(2) + (2)2 = 9 (x + 2)2 = 9 x + 2 = ±3 x + 2 + -2 = -2 ± 3 x = -2 ± 3 x = -2 – 3 x = -5 x = {-5, 1}

3.) x2 – 8x + 16 = 5 3.) x2 – 8x + 16 = 5 x2 – 2(x)(4) + (4)2 = 5 (x – 4)2 = 5 x ≈ {1.8, 6.2}

Solving Quadratic Equations by Completing the Square Section 9 – 3 Day 2

Steps 1.) Find 2.) Find 3.) Add to ax2 + bx

Find the value of c that makes it a perfect square. x2 + 6x + c Step 1: Step 2: (3)2 = 9 Step 3.) c = 9 Check: x2 + 6x + 9 = (x + 3)2

Find the value of c that makes it a perfect square. x2 – 8x + c Step 1: Step 2: (-4)2 = 16 Step 3.) c = 16 Check: x2 – 8x + 16 = (x - 4)2

Solve x2 + 6x + 3 = 10 by completing the square x2 + 6x = 7 Since add 9 to each side x2 + 6x + 9 = 7 + 9 x2 + 2(x)(3) + (3)2 = 16 (x + 3)2 = 16 x + 3 = ±4 x + 3 + -3= -3 ± 4 x = -3 ± 4 x = -3 + 4 x =1 x = -3 - 4 x = -7 x = {-7, 1}

t2 – 4t + 3 = 0 t = 2 + 1 t = 3 t2 – 4t + 3 = 0 t2 – 4t + 3 + -3 = 0 + -3 t2 – 4t = -3 t2 + 2(t)(-2) + (-2)2 = -3 + (-2)2 t2 – 4t + 4 = -3 + 4 (t – 2)2 = 1 t – 2 = ±1 t + -2 + 2 = 2 ±1 t = 2 ±1 t = 2 - 1 t = 1 t = {1,3}

t = {-9,-1} y2 + 10y = -9 t = -5 + 4 t = -1 t = -5 - 4 t = -9 y2 + 10y = -9 y2 + 2(y)(5) + (5)2 = -9 + (5)2 y2 + 10y + 25 = -9 + 25 y2 + 10y + 25 = 16 (t + 5)2 = 16 t + 5 = ±4 t + 5 + -5 = -5 ±4 t = -5 ±4 t = -5 + 4 t = -1 t = -5 - 4 t = -9 t = {-9,-1}