NAME
DATE
4-6
PERIOD
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 ax2 + bx + c = 0. Quadratic Formula
Example
2 -b ± √b - 4ac 2a
The solutions of ax 2 + bx + c = 0, with a ≠ 0, are given by x = − .
Solve x2 - 5x = 14 by using the Quadratic Formula.
Rewrite the equation as x2 - 5x - 14 = 0. -b ± √ b2 - 4ac 2a -(-5) ± √ (-5)2 - 4(1)(-14) = −− 2(1) 5 ± √ 81 = − 2 5±9 = − 2
x= −
Quadratic Formula Replace a with 1, b with -5, and c with -14. Simplify.
= 7 or -2
The solutions are -2 and 7.
Exercises Solve each equation by using the Quadratic Formula.
5, -7 4. 4x2 + 19x - 5 = 0
1 − , -5 4
7. 3x2 + 5x = 2
1 -2, − 3
10. 8x2 + 6x - 9 = 0
3 3 ,− -− 2 4
13. x2 + 6x - 23 = 0
2 -3 ± 4 √
Chapter 4
2. x2 + 10x + 24 = 0
-4, -6
3, 8
5. 14x2 + 9x + 1 = 0
7
2
8. 2y2 + y - 15 = 0
5 − , -3
9. 3x2 - 16x + 16 = 0
4 4, −
2
3
3r 2 11. r2 - − +− =0 5
6. 2x2 - x - 15 = 0
5 3, - −
1 1 -− , -− 2
3. x2 - 11x + 24 = 0
25
2 1 − ,−
12. x2 - 10x - 50 = 0
5 ± 5 √ 3
5 5
14. 4x2 - 12x - 63 = 0
3 ± 6 √2 2
15. x2 - 6x + 21 = 0
3 ± 2i √ 3
−
36
Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. x2 + 2x - 35 = 0
NAME
DATE
4-6
PERIOD
Study Guide and Intervention
(continued)
The Quadratic Formula and the Discriminant
Discriminant
Lesson 4-6
Roots and the Discriminant The expression under the radical sign, b2 - 4ac, in the Quadratic Formula is called the discriminant.
Discriminant
Type and Number of Roots
2
2 rational roots
2
b - 4ac > 0, but not a perfect square
2 irrational roots
b 2 - 4ac = 0
1 rational root
b - 4ac > 0 and a perfect square
2
2 complex roots
b - 4ac < 0
Example
Find the value of the discriminant for each equation. Then describe the number and type of roots for the equation. a. 2x2 + 5x + 3 The discriminant is b2 - 4ac = 52 - 4(2) (3) or 1. The discriminant is a perfect square, so the equation has 2 rational roots.
b. 3x2 - 2x + 5 The discriminant is b2 - 4ac = (-2)2 - 4(3) (5) or -56. The discriminant is negative, so the equation has 2 complex roots.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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. p2 + 12p = -4
2 irrational roots; -6 ± 4 √2 4. x2 + 4x - 4 = 0
2. 9x2 - 6x + 1 = 0
3. 2x2 - 7x - 4 = 0
1 1 rational root; − 3
5. 5x2 - 36x + 7 = 0
1 2 rational roots; - − ,4 2
6. 4x2 - 4x + 11 = 0
2 irrational roots;
2 rational roots;
-160; 2 complex roots;
-2 ± 2 √2
1 − ,7
−
7. x2 - 7x + 6 = 0
2 rational roots; 1, 6 10. 4x2 + 20x + 29 = 0
2 complex roots; 5 -− ±i 2
Chapter 4
1 ± i √ 10 2
5
8. m2 - 8m = -14
2 irrational roots; 2 4 ± √ 11. 6x2 + 26x + 8 = 0
2 rational roots; 1 -4, - − 3
37
9. 25x2 - 40x = -16
4 1 rational root; − 5
12. 4x2 - 4x - 11 = 0
2 irrational roots; 1 − ± √ 3 2
Glencoe Algebra 2
