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

Write each function in standard form. Opener-Same Sheet-9/29 Write each function in standard form. Evaluate b2 – 4ac for the given values of the valuables. 1. 19 = x2 – 8x 2. -61 -24x = 2x2 g(x) = 2x2 + 24x + 61 f(x) = x2 – 8x + 19 4. a = 1, b = 3, c = –3 3. a = 2, b = 7, c = 5 21 9

5-5 Homework Quiz 1. Express in terms of i. Solve each equation. 2. 3x2 + 96 = 0

Objectives Solve quadratic equations using the Quadratic Formula. Classify roots using the discriminant.

Vocabulary discriminant

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.

To subtract fractions, you need a common denominator. Remember!

The symmetry of a quadratic function is evident in the last step, 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.

Songs-Videos 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.

Example 1: Quadratic Functions with Real Zeros Find the zeros of f(x)= 2x2 – 16x + 27 using the Quadratic Formula. Write in simplest form.

Check It Out! Example 1a Find the zeros of f(x) = x2 + 3x – 7 using the Quadratic Formula. x2 + 3x – 7 = 0 x2 – 8x + 10 = 0

Example 2: Quadratic Functions with Complex Zeros Find the zeros of f(x) = 4x2 + 3x + 2 using the Quadratic Formula. f(x)= 4x2 + 3x + 2

Opener-SAME SHEET-9/29 Factor-Need H graph a. 2x2 + 9x + 10 b. 5x2 + 31x + 6

Reteach

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

Make sure the equation is in standard form before you evaluate the discriminant, b2 – 4ac. Caution!

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Opener-SAME SHEET-9/30 Solve by completing the square. t2 – 8t – 5 = 0

Opener-SAME SHEET-9/30 Find the type and member of solutions for each equation. 1. x2 – 14x + 50 2. x2 – 14x + 48 2 distinct nonreal complex 2 distinct real

f(x)=x2 -12x + 36 f(x)=x2 -5x - 6

Opener-SAME SHEET-10/3 Radicals Simpilfy a. √-36 b. √-80 c. 4 √ -12 2. Solve a. 4x2 + 32= 0

5-6 Homework Quiz A Find the zeros of each function by using the Quadratic Formula. 1. f(x) = x2 – 4x – 6

Example 2: Quadratic Functions with Complex Zeros Find the zeros of f(x) = 4x2 + 3x + 2 using the Quadratic Formula. f(x)=x2 -12x + 36

Example 3A: Analyzing Quadratic Equations by Using the Discriminant Find the type and number of solutions for the equation. x2 + 36 = 12x x2 + 40 = 12x two distinct nonreal complex solutions. one distinct real solution.

Example 3C: Analyzing Quadratic Equations by Using the Discriminant Find the type and number of solutions for the equation. x2 + 30 = 12x x2 – 4x = –4 one distinct real solution. two distinct real solutions.

The graph shows related functions 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.

Example 4: Sports Application 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 = –16t2 + 24.6t + 6.5. 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 Continued Step 2 Find the horizontal distance that the shot put will have traveled in this time. x = 29.3t x ≈ 29.3(1.77) Substitute 1.77 for t. x ≈ 51.86 Simplify. x ≈ 52 The shot put will have traveled a horizontal distance of about 52 feet.

Properties of Solving Quadratic Equations

Properties of Solving Quadratic Equations

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No matter which method you use to solve a quadratic equation, you should get the same answer. Helpful Hint

Lesson Quiz: Part I Find the zeros of each function by using the Quadratic Formula. 1. f(x) = 3x2 – 6x – 5 2. g(x) = 2x2 – 6x + 5 Find the type and member of solutions for each equation. 3. x2 – 14x + 50 4. x2 – 14x + 48 2 distinct real 2 distinct nonreal complex

Lesson Quiz: Part II 5. A pebble is tossed from the top of a cliff. The pebble’s height is given by y(t) = –16t2 + 200, 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