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Evaluate the expression.

2. 1. ANSWER. 2. 4. 49. 3. 9. 25. 7. 2. 2. 5. ANSWER. Evaluate the expression. 3. An acorn falls to the ground from a height of 25 feet. How long was the acorn in the air?. 1.25 sec. ANSWER. Evaluate the expression. 25. 5. 2. c =. =. 4. 2. EXAMPLE 1.

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Evaluate the expression.

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  1. 2 1. ANSWER 2 4 49 3 9 25 7 2 2. 5 ANSWER Evaluate the expression.

  2. 3. An acorn falls to the ground from a height of 25 feet. How long was the acorn in the air? 1.25 sec ANSWER Evaluate the expression.

  3. 25 5 2 c = = 4 2 EXAMPLE 1 Complete the square Find the value of cthat makes the expression x2 + 5x + ca perfect square trinomial. Then write the expression as the square of a binomial. STEP 1 Find the value ofc.For the expression to be a perfect squaretrinomial, c needs to be the square of half thecoefficient ofbx. Find the square of half the coefficient of bx.

  4. 5 2 x2 + 5x +c=x2+ 5x + Substitute 25 for c. 4 25 = 4 x 2 + EXAMPLE 1 Complete the square STEP 2 Write the expression as a perfect square trinomial. Then write the expression as the square of a binomial. Square of a binomial

  5. 9 3 4 2 ANSWER ANSWER 16; (x + 4)2 36; (x 6)2 ANSWER ; (x )2 for Example 1 GUIDED PRACTICE Find the value of cthat makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. 1. x2 + 8x + c 2. x2 12x + c 3. x2 + 3x + c

  6. Add , or (–8)2, to each side. –16 2 2 EXAMPLE 2 Solve a quadratic equation Solve x2 – 16x = –15 by completing the square. SOLUTION x2 – 16x = –15 Write original equation. x2 – 16x + (– 8)2=–15 + (– 8)2 (x – 8)2 = –15 + (– 8)2 Write left side as the square of a binomial. (x – 8)2 = 49 Simplify the right side.

  7. ANSWER The solutions of the equation are 8 + 7 = 15 and 8 – 7 = 1. EXAMPLE 2 Standardized Test Practice x – 8 = ±7 Take square roots of each side. x = 8 ± 7 Add 8 to each side.

  8. (15)2– 16(15) –15 (1)2– 16(1) –15 –15 = –15 –15 = –15 ? = ? = EXAMPLE 2 Standardized Test Practice CHECK You can check the solutions in the original equation. If x = 1: If x = 15:

  9. 10 2 , or52, to each side. Add 2 EXAMPLE 3 Solve a quadratic equation in standard form Solve 2x2 + 20x – 8 = 0 by completing the square. SOLUTION 2x2 + 20x – 8 = 0 Write original equation. 2x2 + 20x = 8 Add 8 to each side. x2 + 10x = 4 Divide each side by 2. x2+ 10x + 52= 4 + 52 (x + 5)2 = 29 Write left side as the square of a binomial.

  10. ± x + 5 = 29 ± 29 x = –5 ANSWER The solutions are – 5 + 29 0.39 and – 5 – 29 –10.39. EXAMPLE 3 Solve a quadratic equation in standard form Take square roots of each side. Subtract 5 from each side.

  11. ANSWER 1, 3 ANSWER ANSWER 9.12, 0.88 1.35, 6.65 for Examples 2 and 3 GUIDED PRACTICE 4. x2 – 2x = 3 5. m2 + 10m = –8 6.3g2 – 24g+ 27 = 0

  12. EXAMPLE 4 Solve a multi-step problem CRAFTS You decide to use chalkboard paint to create a chalkboard on a door. You want the chalkboard to have a uniform border as shown. You have enough chalkboard paint to cover 6 square feet. Find the width of the border to the nearest inch.

  13. (7 – 2x) (3 – 2x) 6 = EXAMPLE 4 Solve a multi-step problem SOLUTION STEP 1 Write a verbal model. Then write an equation. Let x be the width (in feet) of the border.

  14. 15 – = x2 – 5x 4 25 5 25 15 25 2 , to each side. x2 –5x + , or Add – – + = 4 2 4 4 4 EXAMPLE 4 Solve a multi-step problem STEP 2 Solve the equation. 6 = (7 – 2x)(3 – 2x) Write equation. 6 = 21 – 20x + 4x2 Multiply binomials. –15 = 4x2 – 20x Subtract 21 from each side. Divide each side by 4.

  15. 5 5 5 5 5 5 5 25 15 – = (x – ) + 2 2 2 2 2 2 2 2 4 4 (x – ) = 2 ± x – = 5 x ± = Add to each side. 2 EXAMPLE 4 Solve a multi-step problem Write right side as the square of a binomial. Simplify left side. Take square roots of each side.

  16. The solutions of the equation are and It is not possible for the width of the border to be 4.08 feet because the width of the door is 3 feet. So, the width of the border is 0.92 foot. Convert 0.92 foot to inches. 5 5 5 5 0.92 4.08 +  2 2 2 2 ANSWER 12 in. 0.92 ft = 11.04 in The width of the border should be about 11 inches. 1 ft EXAMPLE 4 Solve a multi-step problem Multiply by conversion factor

  17. ANSWER The width of the border should be about 13 inches. for Example 4 GUIDED PRACTICE 7. WHAT IF? In Example 4, suppose you have enough chalkboard paint to cover 4 square feet. Find the width of the border to the border to the nearest inch.

  18. –14, 2 ANSWER – 1.29, 9.29 ANSWER Daily Homework Quiz Solve the equation by completing the square. Round to the nearest hundredth, if necessary. 1. x2 + 12x = 28 2.m2 – 8m = 12

  19. 3. What is the width of the border that surrounds this poster? ANSWER 1 in Daily Homework Quiz

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