The Quadratic Formula Quadratic Equation Quadratic Formula

Solve 3x 2 -4x-1=0. Round to the nearesthundredth.

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

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

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. Quick Check x = Write as two equations. – orx = –4 – Approximate. √122 ≈ x – x –4 – or Using the Quadratic Formula LESSON 9-7 Additional Examples

Vertical Motion Object DroppedObject Thrown Upward

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

(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. 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 Quick Check

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 Quick Check

How long will a ball thrown upwards at 20 ft/sec stay in the air if it is thrown from a 100 ft. cliff?