ALGEBRA 1 Lesson 9-7 Warm-Up

ALGEBRA 1 “Using the Quadratic Formula” (9-7) What is the “quadratic formula”? When and how do you use the quadratic formula? Rule: If you complete the square of ax 2 + bx + c = 0, you will derive the quadratic formula (see page 463 for this derivation) which says: You should use the quadratic formula when you cannot factor out “a” before completing the square or you cannot factor the equation. Note: Always write the equation in standard form so you don’t mix up the a, b, and c when “plugging them into” (substituting) the quadratic formula. Example: Solve 2x = -4x 2x x = -4x + 4x Add 4x from both side and write the equation in standard from 2x 2 + 4x - 7 = 0a = 2, b = 4, c = -7 Use the quadratic formula Substitute (a = 2, b = 4, c = -7)

ALGEBRA 1 “Using the Quadratic Formula” (9-7) Simplify Since  means “+” or “-”, write this as twoequations Check: Substitute 1.12 for x Substitute for x 2(1.12)  -4(1.12) 2(-3.12)  -4(-3.12) 2(1.2544) – 7  (9.7344) – 7  – 7  ) – 7     

ALGEBRA 1 Solve x = –3x. x 2 + 3x + 2 = 0Add 3x to each side and write in standard form. x = –b ± b 2 – 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 = – x = –3 – 1 2 or Write as two equations. x = –1orx = –2Simplify. Using the Quadratic Formula LESSON 9-7 Additional Examples

ALGEBRA 1 (continued) Check: (–1) 2 + 3(–1) + 2 0(–2) 2 + 3(–2) – – = 0 for x = –2for x = –1 Using the Quadratic Formula LESSON 9-7 Additional Examples

ALGEBRA 1 Solve 3x 2 + 4x – 8 = 0. Round the solutions to the nearest hundredth. x = –b ± b 2 – 4ac 2a Use the quadratic formula.x = –4 ± 4 2 – 4(3)(–8) 2(3) Substitute 3 for a, 4 for b, and –8 for c. –4 ± x = x 1.10orx –2.43 Simplify. Round to the nearest hundredth. x = Write as two equations. – orx = –4 – Approximate. √122 ≈ x – x –4 – or Using the Quadratic Formula LESSON 9-7 Additional Examples

ALGEBRA 1 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, how long will it be in the air? Round to the nearest hundredth of a second. Step 1: Use the vertical motion formula. Step 2: Use the quadratic formula. x = –b ± b 2 – 4ac 2a h = –16t 2 + vt + c 0 = –16t t + 2 Substitute 0 for h, 15 for v, and 2 for c. Using the Quadratic Formula LESSON 9-7 Additional Examples

ALGEBRA 1 (continued) t = –15 ± 15 2 – 4(–16)(2) 2(–16) Substitute –16 for a, 15 for b, 2 for c, and t for x. t = – –32 or t = –15 – –32 Write as two equations. t–0.12ort1.06Simplify. Use the positive answer because it is the only reasonable answer in this situation. Since the solution can’t be negative, the ball will land in about 1.06 seconds. –15 ± –32 t = Simplify. –15 ± 353 –32 t = Using the Quadratic Formula LESSON 9-7 Additional Examples

ALGEBRA 1 “Using the Quadratic Formula” (9-7) How can you tell which method to use when solving a quadratic equation? Tip: You can use the quadratic formula to solve any quadratic equation, but sometimes there are much easier methods. Use the following table to help you decide which method to use when solving a quadratic equation.

ALGEBRA 1 Which method(s) would you choose to solve each equation? Justify your reasoning. a. 5x 2 + 8x – 14 = 0Quadratic formula; the equation cannot be factored easily. b. 25x 2 – 169 = 0 Square roots; there is no x term. c. x 2 – 2x – 3 = 0 Factoring; the equation is easily factorable. d. x 2 – 5x + 3 = 0Quadratic formula, completing the square, or graphing; the x 2 term is 1, but the equation is not factorable. e. 16x 2 – 96x = 0Quadratic formula; the equation cannot be factored easily and the numbers are large. Using the Quadratic Formula LESSON 9-7 Additional Examples

ALGEBRA 1 1.Solve 2x 2 – 11x + 12 = 0 by using the quadratic formula. 2.Solve 4x 2 – 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, seconds Using the Quadratic Formula LESSON 9-7 Lesson Quiz