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

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

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

  2. Vocabulary discriminant

  3. 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.

  4. Example 1: Quadratic Functions with Real Zeros Find the zeros of f(x)= 2x2 – 16x + 27 using the Quadratic Formula. 2x2– 16x + 27 = 0 Set f(x) = 0. Write the Quadratic Formula. Substitute 2 for a, –16 for b, and 27 for c. Simplify. Write in simplest form.

  5. Check It Out! Example 1a Find the zeros of f(x) = x2 + 3x –7 using the Quadratic Formula. x2 + 3x–7 = 0 Set f(x) = 0. Write the Quadratic Formula. Substitute 1 for a, 3 for b, and –7 for c. Simplify. Write in simplest form.

  6. Check It Out! Example 1b Find the zeros of f(x)= x2 – 8x + 10 using the Quadratic Formula. x2 – 8x + 10 = 0 Set f(x) = 0. Write the Quadratic Formula. Substitute 1 for a, –8 for b, and 10 for c. Simplify. Write in simplest form.

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

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

  9. Example 3A: Analyzing Quadratic Equations by Using the Discriminant Find the type and number of solutions for the equation. x2 + 36 = 12x x2–12x + 36 = 0 b2 – 4ac (–12)2 – 4(1)(36) 144 – 144 = 0 b2 – 4ac = 0 The equation has one distinct real solution.

  10. Example 3B: Analyzing Quadratic Equations by Using the Discriminant Find the type and number of solutions for the equation. x2 + 40 = 12x x2–12x + 40 = 0 b2 – 4ac (–12)2 – 4(1)(40) 144 – 160 = –16 b2 –4ac < 0 The equation has two distinct nonreal complex solutions.

  11. Example 3C: Analyzing Quadratic Equations by Using the Discriminant Find the type and number of solutions for the equation. x2 + 30 = 12x x2–12x + 30 = 0 b2 – 4ac (–12)2 – 4(1)(30) 144 – 120 = 24 b2 – 4ac > 0 The equation has two distinct real solutions.

  12. 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

  13. 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

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