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Holt McDougal Algebra The Quadratic Formula 5-6 The Quadratic Formula Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Algebra 2

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5-6 The Quadratic Formula Warm Up Write each function in standard form. Evaluate b 2 – 4ac for the given values of the valuables. 1. f(x) = (x – 4) g(x) = 2(x + 6) 2 – 11 f(x) = x 2 – 8x + 19 g(x) = 2x x a = 1, b = 3, c = –3 3. a = 2, b = 7, c =

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Holt McDougal Algebra The Quadratic Formula Solve quadratic equations using the Quadratic Formula. Classify roots using the discriminant. Objectives

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Holt McDougal Algebra The Quadratic Formula discriminant Vocabulary

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Holt McDougal Algebra The Quadratic Formula You have learned several methods for solving quadratic equations: graphing, making tables, factoring, using square roots, and completing the square. Another method is to use the Quadratic Formula, which allows you to solve a quadratic equation in standard form. By completing the square on the standard form of a quadratic equation, you can determine the Quadratic Formula.

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Holt McDougal Algebra The Quadratic Formula

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Holt McDougal Algebra The Quadratic Formula To subtract fractions, you need a common denominator. Remember!

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Holt McDougal Algebra The Quadratic Formula The symmetry of a quadratic function is evident in the last step,. These two zeros are the same distance,, away from the axis of symmetry,,with one zero on either side of the vertex.

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Holt McDougal Algebra The Quadratic Formula You can use the Quadratic Formula to solve any quadratic equation that is written in standard form, including equations with real solutions or complex solutions.

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Holt McDougal Algebra The Quadratic Formula Find the zeros of f(x)= 2x 2 – 16x + 27 using the Quadratic Formula. Example 1: Quadratic Functions with Real Zeros Set f(x) = 0. Write the Quadratic Formula. Substitute 2 for a, –16 for b, and 27 for c. Simplify. Write in simplest form. 2x 2 – 16x + 27 = 0

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Holt McDougal Algebra The Quadratic Formula Check Solve by completing the square. Example 1 Continued

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Holt McDougal Algebra The Quadratic Formula Find the zeros of f(x) = x 2 + 3x – 7 using the Quadratic Formula. Set f(x) = 0. Write the Quadratic Formula. Substitute 1 for a, 3 for b, and –7 for c. Simplify. Write in simplest form. Check It Out! Example 1a x 2 + 3x – 7 = 0

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Holt McDougal Algebra The Quadratic Formula Check Solve by completing the square. x 2 + 3x – 7 = 0 x 2 + 3x = 7 Check It Out! Example 1a Continued

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Holt McDougal Algebra The Quadratic Formula Find the zeros of f(x)= x 2 – 8x + 10 using the Quadratic Formula. Set f(x) = 0. Write the Quadratic Formula. Substitute 1 for a, –8 for b, and 10 for c. Simplify. Write in simplest form. Check It Out! Example 1b x 2 – 8x + 10 = 0

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Holt McDougal Algebra The Quadratic Formula Check Solve by completing the square. x 2 – 8x + 10 = 0 x 2 – 8x = –10 x 2 – 8x + 16 = – (x + 4) 2 = 6 Check It Out! Example 1b Continued

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Holt McDougal Algebra The Quadratic Formula Find the zeros of f(x) = 4x 2 + 3x + 2 using the Quadratic Formula. Example 2: Quadratic Functions with Complex Zeros Set f(x) = 0. Write the Quadratic Formula. Substitute 4 for a, 3 for b, and 2 for c. Simplify. Write in terms of i. f(x)= 4x 2 + 3x + 2

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Holt McDougal Algebra The Quadratic Formula Find the zeros of g(x) = 3x 2 – x + 8 using the Quadratic Formula. Set f(x) = 0 Write the Quadratic Formula. Substitute 3 for a, –1 for b, and 8 for c. Simplify. Write in terms of i. Check It Out! Example 2

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Holt McDougal Algebra The Quadratic Formula The discriminant is part of the Quadratic Formula that you can use to determine the number of real roots of a quadratic equation.

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Holt McDougal Algebra The Quadratic Formula Make sure the equation is in standard form before you evaluate the discriminant, b 2 – 4ac. Caution!

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Holt McDougal Algebra The Quadratic Formula Find the type and number of solutions for the equation. Example 3A: Analyzing Quadratic Equations by Using the Discriminant x = 12x x 2 – 12x + 36 = 0 b 2 – 4ac (–12) 2 – 4(1)(36) 144 – 144 = 0 b 2 – 4ac = 0 The equation has one distinct real solution.

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Holt McDougal Algebra The Quadratic Formula Find the type and number of solutions for the equation. Example 3B: Analyzing Quadratic Equations by Using the Discriminant x = 12x x 2 – 12x + 40 = 0 b 2 – 4ac (–12) 2 – 4(1)(40) 144 – 160 = –16 b 2 –4ac < 0 The equation has two distinct nonreal complex solutions.

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Holt McDougal Algebra The Quadratic Formula Find the type and number of solutions for the equation. Example 3C: Analyzing Quadratic Equations by Using the Discriminant x = 12x x 2 – 12x + 30 = 0 b 2 – 4ac (–12) 2 – 4(1)(30) 144 – 120 = 24 b 2 – 4ac > 0 The equation has two distinct real solutions.

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Holt McDougal Algebra The Quadratic Formula Check It Out! Example 3a Find the type and number of solutions for the equation. x 2 – 4x = –4 x 2 – 4x + 4 = 0 b 2 – 4ac (–4) 2 – 4(1)(4) 16 – 16 = 0 b 2 – 4ac = 0 The equation has one distinct real solution.

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Holt McDougal Algebra The Quadratic Formula Check It Out! Example 3b Find the type and number of solutions for the equation. x 2 – 4x = –8 x 2 – 4x + 8 = 0 b 2 – 4ac (–4) 2 – 4(1)(8) 16 – 32 = –16 b 2 – 4ac < 0 The equation has two distinct nonreal complex solutions.

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Holt McDougal Algebra The Quadratic Formula Check It Out! Example 3c Find the type and number of solutions for each equation. x 2 – 4x = 2 x 2 – 4x – 2 = 0 b 2 – 4ac (–4) 2 – 4(1)(–2) = 24 b 2 – 4ac > 0 The equation has two distinct real solutions.

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Holt McDougal Algebra The Quadratic Formula The graph shows related functions. Notice that the number of real solutions for the equation can be changed by changing the value of the constant c.

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Holt McDougal Algebra The Quadratic Formula An athlete on a track team throws a shot put. The height y of the shot put in feet t seconds after it is thrown is modeled by y = –16t t The horizontal distance x in between the athlete and the shot put is modeled by x = 29.3t. To the nearest foot, how far does the shot put land from the athlete? Example 4: Sports Application

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Holt McDougal Algebra The Quadratic Formula Step 1 Use the first equation to determine how long it will take the shot put to hit the ground. Set the height of the shot put equal to 0 feet, and the use the quadratic formula to solve for t. y = –16t t Example 4 Continued Set y equal to 0. 0 = –16t t Use the Quadratic Formula. Substitute –16 for a, 24.6 for b, and 6.5 for c.

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Holt McDougal Algebra The Quadratic Formula Example 4 Continued Simplify. The time cannot be negative, so the shot put hits the ground about 1.8 seconds after it is released.

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Holt McDougal Algebra The Quadratic Formula Step 2 Find the horizontal distance that the shot put will have traveled in this time. x = 29.3t Example 4 Continued Substitute 1.77 for t. Simplify. x 29.3(1.77) x x 52 The shot put will have traveled a horizontal distance of about 52 feet.

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Holt McDougal Algebra The Quadratic Formula Check Use substitution to check that the shot put hits the ground after about 1.77 seconds. Example 4 Continued The height is approximately equal to 0 when t = y = –16t t y –16(1.77) (1.77) y – y –0.09

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Holt McDougal Algebra The Quadratic Formula Check It Out! Example 4 The pilots altitude decreases, which changes the function describing the waters height to y = –16t 2 –2t To the nearest foot, at what horizontal distance from the target should the pilot begin releasing the water? A pilot of a helicopter plans to release a bucket of water on a forest fire. The height y in feet of the water t seconds after its release is modeled by y = –16t 2 – 2t the horizontal distance x in feet between the water and its point of release is modeled by x = 91t.

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Holt McDougal Algebra The Quadratic Formula Step 1 Use the equation to determine how long it will take the water to hit the ground. Set the height of the water equal to 0 feet, and then use the quadratic formula for t. y = –16t 2 – 2t Set y equal to 0. 0 = –16t 2 – 2t Write the Quadratic Formula. Substitute –16 for a, –2 for b, and 400 for c. Check It Out! Example 4 Continued

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Holt McDougal Algebra The Quadratic Formula Check It Out! Example 4 Simplify. t –5.063 or t The time cannot be negative, so the water lands on a target about seconds after it is released.

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Holt McDougal Algebra The Quadratic Formula Check It Out! Example 4 Continued Simplify. Step 2 The horizontal distance x in feet between the water and its point of release is modeled by x = 91t. Find the horizontal distance that the water will have traveled in this time. x = 91t Substitute for t. x 91(4.937) x x 449 The water will have traveled a horizontal distance of about 449 feet. Therefore, the pilot should start releasing the water when the horizontal distance between the helicopter and the fire is 449 feet.

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Holt McDougal Algebra The Quadratic Formula Check It Out! Example 4 Continued Check Use substitution to check that the water hits the ground after about seconds. The height is approximately equal to 0 when t = y = –16t 2 – 2t y –16(4.937) 2 – 2(4.937) y – – y 0.143

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Holt McDougal Algebra The Quadratic Formula Properties of Solving Quadratic Equations

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Holt McDougal Algebra The Quadratic Formula Properties of Solving Quadratic Equations

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Holt McDougal Algebra The Quadratic Formula No matter which method you use to solve a quadratic equation, you should get the same answer. Helpful Hint

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Holt McDougal Algebra The Quadratic Formula Lesson Quiz: Part I Find the zeros of each function by using the Quadratic Formula. 1. f(x) = 3x 2 – 6x – 5 2. g(x) = 2x 2 – 6x + 5 Find the type and member of solutions for each equation. 3. x 2 – 14x x 2 – 14x distinct nonreal complex 2 distinct real

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Holt McDougal Algebra The Quadratic Formula Lesson Quiz: Part II 5. A pebble is tossed from the top of a cliff. The pebbles height is given by y(t) = –16t , where t is the time in seconds. Its horizontal distance in feet from the base of the cliff is given by d(t) = 5t. How far will the pebble be from the base of the cliff when it hits the ground? about 18 ft

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