Using the Quadratic Formula ALGEBRA 1 LESSON 9-6 Solve x2 + 2 = –3x using the quadratic formula. x2 + 3x + 2 = 0 Add 3x to each side and write in standard form. x = –b ± b2 – 4ac 2a Use the quadratic formula. x = –3 ± (–3)2 – 4(1)(2) 2(1) Substitute 1 for a, 3 for b, and 2 for c. x = –3 ± 1 2 Simplify. x = –3 + 1 2 –3 – 1 or Write two solutions. x = –1 or x = –2 Simplify. 9-6

Using the Quadratic Formula ALGEBRA 1 LESSON 9-6 (continued) Check: for x = –1 for x = –2 (–1)2 + 3(–1) + 2 0 (–2)2 + 3(–2) + 2 0 1 – 3 + 2 0 4 – 6 + 2 0 0 = 0 9-6

Using the Quadratic Formula ALGEBRA 1 LESSON 9-6 Solve 3x2 + 4x – 8 = 0. Round the solutions to the nearest hundredth. x = –b ± b2 – 4ac 2a Use the quadratic formula. x = –4 ± 42 – 4(3)(–8) 2(3) Substitute 3 for a, 4 for b, and –8 for c. –4 ± 112 6 x = or Write two solutions. –4 + 112 6 –4 – 112 Use a calculator. x –4 + 10.583005244 6 –4 – 10.583005244 or x 1.10 x –2.43 Round to the nearest hundredth. 9-6

Using the Quadratic Formula ALGEBRA 1 LESSON 9-6 A child throws a ball upward with an initial upward velocity of 15 ft/s from a height of 2 ft. If no one catches the ball, after how many seconds will it land? Use the vertical motion formula h = –16t2 + vt + c, where h = 0, v = velocity, c = starting height, and t = time to land. Round to the nearest hundredth of a second. Step 1:  Use the vertical motion formula. h = –16t2 + vt + c 0 = –16t2 + 15t + 2  Substitute 0 for h, 15 for v, and 2 for c. Step 2: Use the quadratic formula. x = –b ± b2 – 4ac 2a 9-6

Using the Quadratic Formula ALGEBRA 1 LESSON 9-6 (continued) t = –15 ± 152 – 4(–16)(2) 2(–16) Substitute –16 for a, 15 for b, 2 for c, and t for x. –15 ± 225 + 128 –32 t = Simplify. –15 ± 353 t = –15 + 18.79 –32 or –15 – 18.79 Write two solutions. t –0.12 or 1.06 Simplify. Use the positive answer because it is the only reasonable answer in this situation. The ball will land in about 1.06 seconds. 9-6

Using the Quadratic Formula ALGEBRA 1 LESSON 9-6 Which method(s) would you choose to solve each equation? Justify your reasoning. a. 5x2 + 8x – 14 = 0 Quadratic formula; the equation cannot be factored easily. b. 25x2 – 169 = 0 Square roots; there is no x term. c. x2 – 2x – 3 = 0 Factoring; the equation is easily factorable. d. x2 – 5x + 3 = 0 Quadratic formula, completing the square, or graphing; the x2 term is 1, but the equation is not factorable. e. 16x2 – 96x + 135 = 0 Quadratic formula; the equation cannot be factored easily and the numbers are large. 9-6

Using the Discriminant ALGEBRA 1 LESSON 9-6 Find the number of solutions of x2 = –3x – 7 using the discriminant. x2 + 3x + 7 = 0 Write in standard form. b2 – 4ac = 32 – 4(1)(7) Evaluate the discriminant. Substitute for a, b, and c. = 9 – 28 Use the order of operations. = –19 Simplify. Since –19 < 0, the equation has no solution. 9-6

Using the Discriminant ALGEBRA 1 LESSON 9-6 A football is kicked from a starting height of 3 ft with an initial upward velocity of 40 ft/s. Will the football ever reach a height of 30 ft? h = –16t2 + vt + c Use the vertical motion formula. 30 = –16t2 + 40t + 3 Substitute 30 for h, 40 for v, and 3 for c. 0 = –16t2 + 40t – 27 Write in standard form. b2 – 4ac = (40)2 – 4 (–16)(–27) Evaluate the discriminant. = 1600 – 1728 Use the order of operations. = –128 Simplify. The discriminant is negative. The football will never reach a height of 30 ft. 9-6

Using the Quadratic Formula ALGEBRA 1 LESSON 9-6 1. Solve 2x2 – 11x + 12 = 0 by using the quadratic formula. 2. Solve 4x2 – 12x = 64. Round the solutions to the nearest hundredth. 3. Suppose a model rocket is launched from a platform 2 ft above the ground with an initial upward velocity of 100 ft/s. After how many seconds will the rocket hit the ground? Round the solution to the nearest hundredth. 1.5, 4 –2.77, 5.77 6.27 seconds 9-6

Using the Discriminant ALGEBRA 1 LESSON 9-6 Find the number of solutions for each equation. 1. 3x2 – 4x = 7 2. 4x2 = 4x – 1 3. –3x2 + 2x – 12 = 0 4. A ball is thrown from a starting height of 4 ft with an initial upward velocity of 30 ft/s. Is it possible for the ball to reach a height of 18 ft? two one none yes 9-6