Do Now: Pass out calculators. Write down your assignments for the week for a scholar dollar. Use square roots to solve the quadratic equations below: 3x2=27 P2 + 12 = 12 a2 +3 = -4 4(x +6)2 = 32 20 = 2(m +5)2 Answers: -3, 3 No solution -3.17, -8.83 -8.16, -1.84

Objective: To solve quadratic equations by completing the square.

Completing the Square: For an expression of the form: x2 + bx, you can add a constant c to the expression so that the expression x2 + bx + c is a perfect square trinomial. *This process is called completing the square.

Completing the Square: To complete the square for the expression x2 + bx .. Add the square root of half the coefficient of the term bx. So.. x2 + bx + (b/2)2 = (x + b/2)2

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

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. x2 + 5x + c = x2 + 5x + 25 4 Substitute 25 for c. 4 5 2 + x = Square of a binomial

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

Solve a quadratic equation EXAMPLE 2 Solve a quadratic equation Solve x2 – 16x = –15 by completing the square. SOLUTION x2 – 16x = –15 Write original equation. Add , or (– 8)2, to each side. – 16 2 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.

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

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

Solve a quadratic equation in standard form 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. Add 10 2 , or 52, to each side. x2 + 10x + 52 = 4 + 52 (x + 5)2 = 29 Write left side as the square of a binomial.

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

GUIDED PRACTICE for Examples 2 and 3 4. x2 – 2x = 3 ANSWER 1, 3 5. m2 + 10m = –8 ANSWER 9.12, 0.88 6. 3g2 – 24g + 27 = 0 ANSWER 1.35, 6.65

Exit Ticket: Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. x2 - 3x + c Complete the square. 2. x2 + 10x = 24 3. 2x2 + 24x + 10 = 0 Answers: 1. 9/4; (x - 3/2)2 2. -12, 2 3. -11.57, -0.43

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.

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. 6 = (7 – 2x) (3 – 2x)

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. – = x2 – 5x 15 4 Divide each side by 4. 25 4 – 15 + = x2 – 5x + Add – 5 2 , or 25 4 , to each side.

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

EXAMPLE 4 Solve a multi-step problem 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 2 + 4.08  0.92 0.92 ft 12 in. 1 ft = 11.04 in Multiply by conversion factor ANSWER The width of the border should be about 11 inches.