Section 5.7 Completing the Square

Simplify each expression. Completing the Square ALGEBRA 2 LESSON 5-7 (For help, go to Lessons 5-1 and page 262.) 9 16 1. (x – 3)(x – 3) 2. (2x – 1)(2x – 1) 3. (x + 4)(x + 4) – 3 4. ± 25 5. ± 48 6. ± –4 7. ± Simplify each expression. Check Skills You’ll Need 5-7

1. (x – 3)(x – 3) = x2 – 2(3)x + 32 = x2 – 6x + 9 Completing the Square ALGEBRA 2 LESSON 5-7 Solutions 9 16 3 4 1. (x – 3)(x – 3) = x2 – 2(3)x + 32 = x2 – 6x + 9 2. (2x – 1)(2x – 1) = (2x)2 – 2(1)(2x) + 12 = 4x2 – 4x + 1 3. (x + 4)(x + 4) – 3 = x2 + 2(4)x + 42 – 3 = x2 + 8x + 16 – 3 = x2 + 8x + 13 4. ± 25 = ±5 5. ± 48 = ± 16 • 3 = ±4 3 6. ± –4 = ± –1 • 4 = ±2i 7. ± = ± = ± 5-7

(x – 6)2 = 9 Factor the trinomial. Completing the Square ALGEBRA 2 LESSON 5-7 Solve x2 – 12x + 36 = 9. x2 – 12x + 36 = 9 (x – 6)2 = 9 Factor the trinomial. x – 6 = ±3 Find the square root of each side. x – 6 = 3 or x – 6 = –3 Solve for x. x = 9 or x = 3 Quick Check 5-7

Find the missing value to complete the square: x2 + 20x + . Completing the Square ALGEBRA 2 LESSON 5-7 Find the missing value to complete the square: x2 + 20x + . = = 100 Find . Substitute 20 for b. b 2 20 x2 + 20x + 100 Complete the square. Quick Check 5-7

x2 + 4x = –1 Rewrite so all terms containing x are on one side. Completing the Square ALGEBRA 2 LESSON 5-7 Solve x2 + 4x + 1 = 0. 4 2 = 4 Find . 2 b 2 2 x2 + 4x = –1 Rewrite so all terms containing x are on one side. x2 + 4x + 4 = –1 + 4 Complete the square by adding 4 to each side. (x + 2)2 = 3 Factor the perfect square trinomal. x + 2 = ± 3 Find the square root of each side. x = –2 ± 3 Solve for x. 5-7

Completing the Square (continued) Check: ALGEBRA 2 LESSON 5-7 (continued) Check: (–2)2 – 2(–2 3) + ( 3)2 – 8 – 4 3 + 1 (–2 + 3)2 + 4(–2 + 3) + 1 x2 + 4x + 1 0 = 0 4 – 4 3 + 3 – 8 + 4 3 + 1 (4 + 3 – 8 + 1) + (–4 3 + 4 3) (–2)2 + 2(–2 3) + ( 3)2 + (–8) + 4 3 + 1 (–2 – 3)2 + 4(–2 – 3) + 1 4 + 4 3 + 3 – 8 – 4 3 + 1 (4 + 3 – 8 + 1) + (4 3 – 4 3) Quick Check 5-7

Rewrite so all terms containing x are on one side. Completing the Square ALGEBRA 2 LESSON 5-7 Solve x2 + 6x + 12 = 0. 6 2 2 b 2 2 = 9 Find . x2 + 6x = –12 Rewrite so all terms containing x are on one side. x2 + 6x + 9 = –12 + 9 Complete the square by adding 9 to each side. (x + 3)2 = –3 Factor the perfect square trinomial. x + 3 = ± –3 Find the square root of each side. x = –3 ± –3 Solve for x. = –3 ± i 3 Simplify. Quick Check 5-7

= Completing the Square Solve 2x2 + 7x – 1 = 0. x2 + x – = 0 ALGEBRA 2 LESSON 5-7 Solve 2x2 + 7x – 1 = 0. x2 + x – = 0 7 2 1 Divide each side by 2. x2 + x = 1 2 7 Rewrite so all terms containing x are on one side. = 7 2 49 16 Find . b x2 + x + = + 7 2 49 16 1 Complete the square by adding to each side. x + = 7 4 57 16 Factor the perfect square trinomial. 2 7 4 57 x + = ± Find the square root of each side. x = – ± 7 4 Solve for x. 57 Quick Check 5-7

Write y = x2 + 5x + 2 in vertex form. Completing the Square ALGEBRA 2 LESSON 5-7 Write y = x2 + 5x + 2 in vertex form. y = x2 + 5x + 2 = x2 + 5x + + 2 – 5 2 Complete the square. Add and subtract on the right side. = x + + 2 – 25 4 5 2 Factor the perfect square trinomial. = x + – 5 2 17 4 Simplify. The vertex form is y = x + – . 5 2 17 4 Quick Check 5-7

Factor –1 from the first two terms. Completing the Square ALGEBRA 2 LESSON 5-7 Quick Check The monthly profit P from the sales of rugs woven by a family rug-making business depends on the price r that they charge for a rug. The profit is model by P = –r 2 + 500r – 59,500. Write the function in vertex form. Use the vertex form to find the price that yields the maximum monthly profit and the amount of the maximum profit. P = –r 2 + 500r – 59500 = –(r 2 – 500r) – 59500 Factor –1 from the first two terms. = –[r 2 – 500r + (–250)2] – 59500 + (–250)2 Add and subtract (–250)2 on the right side. = –(r – 250)2 – 59500 + 62500 Factor the perfect square trinomial. = –(r – 250)2 + 3000 Simplify in vertex form. The vertex is (250, 3000). A price of $250 per rug gives a maximum monthly profit of $3000. 5-7

Simplify each expression. 3. x2 – 6x – 16 = 0 4. x2 – 14x + 74 = 0 Completing the Square ALGEBRA 2 LESSON 5-7 Complete the square. 1. x2 + 60x + 2. x2 – 7x + Simplify each expression. 3. x2 – 6x – 16 = 0 4. x2 – 14x + 74 = 0 5. 3x2 + 5x – 28 = 0 6. 4x2 – 6x + 3 = 0 49 9 900 –2, 8 7 ± 5i –4, 7 3 3 4 ± i 5-7