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The materials are organized by chapter and lesson with two Study Guide and Intervention worksheets for every lesson in Glencoe Algebra 2. Always keep your

Study Guide and Intervention

2. 6 4 2. 4 6 1. 16 23. 4. 1 22. 1. 4. Page 2. © Glencoe/McGraw-Hill. 2. Glencoe Algebra 2. Formulas A formula is a mathematical sentence that uses variables to ...

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organized by chapter and lesson with two Study Guide and Intervention worksheets for every lesson in Glencoe Algebra 2. Always keep your workbook handy.

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4-6

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Chapter 4 36 Glencoe Algebra 2

Study Guide and Intervention

The Quadratic Formula and the Discriminant

Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form ax 2 + bx + c = 0.Quadratic FormulaThe solutions of ax 2 + bx + c = 0, with a ≠ 0, are given by x = b

± ⎷ ???? b

2 - 4ac 2 a

Solve x

2 - 5x = 14 by using the Quadratic Formula.

Rewrite the equation as

x 2 - 5x - 14 = 0. x b b 2 - 4ac ? 2 a

Quadratic Formula

5) ± ⎷

(-5) 2 - 4(1)(-14) 2(1)

Replace a with 1, b with -5, and c with -14.

5 ± ⎷ ?? 81

Simplify.

5 ± 9

2 = 7 or -2

The solutions are

2 and 7.

Exercises

Solve each equation by using the Quadratic Formula. 1. x 2 + 2x - 35 = 0 2. x 2 + 10x + 24 = 0 3. x 2 - 11x + 24 = 0 4. 4x 2 + 19x - 5 = 0 5. 14x 2 + 9x + 1 = 0 6. 2x 2 - x - 15 = 0 7. 3x 2 + 5x = 2 8. 2y

2 + y - 15 = 0 9. 3x

2 - 16x + 16 = 0

10. 8x

2 + 6x - 9 = 0 11. r 2 3 r 5 2 25
= 0 12. x 2 - 10x - 50 = 0 13. x 2 + 6x - 23 = 0 14. 4x 2 - 12x - 63 = 0 15. x2 - 6x + 21 = 0

Example

5, -7 -4, -6 3, 8

1 4 , -5 - 1 2 1 7 3, - 5 2 -2, 1 ? 3 5 2 , -3 4, 4 3 3 2 3 4 2 5 1 5

5 ± 5

3 -3 ± 4 2

3 ± 6

2 2

3 ± 2i

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Lesson 4-6

4-6

PDF Pass

Chapter 4 37 Glencoe Algebra 2

Roots and the Discriminant

DiscriminantThe expression under the radical sign, b 2 - 4ac, in the Quadratic Formula is called the discriminant.

DiscriminantType and Number of Roots

b 2 - 4ac > 0 and a perfect square 2 rational roots b 2 - 4ac > 0, but not a perfect square2 irrational roots b 2 - 4ac = 01 rational root b 2 - 4ac < 02 complex roots Find the value of the discriminant for each equation. Then describe the number and type of roots for the equation. a. 2 x 2 + 5x + 3

The discriminant is

b 2 - 4ac = 5 2 - 4(2) (3) or 1.

The discriminant is a perfect square, so

the equation has 2 rational roots.b. 3x 2 - 2x + 5

The discriminant is

b 2 - 4ac = (-2) 2 - 4(3) (5) or -56.

The discriminant is negative, so the

equation has 2 complex roots.

Exercises

Complete parts a

c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. 1. p 2 + 12p = -4 2. 9x 2 - 6x + 1 = 0 3. 2x 2 - 7x - 4 = 0 4. x 2 + 4x - 4 = 0 5. 5x 2 - 36x + 7 = 0 6. 4x 2 - 4x + 11 = 0 7. x 2 - 7x + 6 = 0 8. m 2 - 8m = -14 9. 25x 2 - 40x = -16

10. 4x

2 + 20x + 29 = 0 11. 6x 2 + 26x + 8 = 0 12. 4x 2 - 4x - 11 = 0

Example

Study Guide and Intervention (continued)

The Quadratic Formula and the Discriminant

2 irrational roots; 1 rational root;

1 3

2 rational roots; -

1 2 , 4 -6 ± 4 2

2 irrational roots; 2 rational roots; -160; 2 complex roots;

-2 ± 2 2 1 5 , 7

1 ± i

10 2

2 rational roots; 2 irrational roots; 1 rational root;

4 5

1, 6 4 ±

2 complex roots; 2 rational roots; 2 irrational roots;

5 2

± i -4, -

1 3 1 2 3

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[PDF] 4-6 - Study Guide and Intervention

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Chapter 4 36 Glencoe Algebra 2

Study Guide and Intervention

The Quadratic Formula and the Discriminant

± ⎷ ???? b

Solve x

Rewrite the equation as

Quadratic Formula

5) ± ⎷

Replace a with 1, b with -5, and c with -14.

5 ± ⎷ ?? 81

Simplify.

5 ± 9

The solutions are

2 and 7.

Exercises

2 + y - 15 = 0 9. 3x

10. 8x

Example

5, -7 -4, -6 3, 8

5 ± 5

3 ± 6

3 ± 2i

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Lesson 4-6

PDF Pass

Chapter 4 37 Glencoe Algebra 2

Roots and the Discriminant

DiscriminantType and Number of Roots

The discriminant is

The discriminant is a perfect square, so

The discriminant is

The discriminant is negative, so the

Exercises

Complete parts a

10. 4x

Example

Study Guide and Intervention (continued)

The Quadratic Formula and the Discriminant

2 irrational roots; 1 rational root;

2 rational roots; -

2 irrational roots; 2 rational roots; -160; 2 complex roots;

1 ± i

2 rational roots; 2 irrational roots; 1 rational root;

1, 6 4 ±

2 complex roots; 2 rational roots; 2 irrational roots;

± i -4, -

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