10-6 The Quadratic Formula and the Discriminant
Name
Date
Use the discriminant to describe the root(s) of each equation.
⫺ 8x ⫹ 2 ⫽ 0
a ⫽ 1, b ⫽ ⫺8, and c ⫽ 2
Identify a, b, and c.
2
2
b ⫺ 4ac ⫽ (⫺8) ⫺ 4(1)(2) ⫽ 64 ⫺ 8 ⫽ 56
Substitute into b2 ⫺ 4ac
x2
and simplify.
The discriminant is 56, which is positive but not a perfect square.
The equation has two irrational roots.
Remember: If the discriminant is positive and a perfect square,
the equation will have two rational roots.
y
6
3
x
2 0
3
6
9
2 4 6 8 10
12
y
x2 ⫺ 8x ⫹ 16 ⫽ 0
a ⫽ 1, b ⫽ ⫺8, and c ⫽ 16
Identify a, b, and c.
2
2
b ⫺ 4ac ⫽ (⫺8) ⫺ 4(1)(16) ⫽ 64 ⫺ 64 ⫽ 0
Substitute into b2 ⫺ 4ac
and simplify.
12
9
6
3
x
2 0
3
6
2 4 6 8 10
The discriminant is 0. There is one rational root.
x2 ⫺ 8x ⫹ 17 ⫽ 0
a ⫽ 1, b ⫽ ⫺8, and c ⫽ 17
Identify a, b, and c.
b2 ⫺ 4ac ⫽ (⫺8)2 ⫺ 4(1)(17) ⫽ 64 ⫺ 68 ⫽ ⫺4
y
Substitute into b2 ⫺ 4ac
and simplify.
The discriminant is ⫺4, which is less than zero.
There are no real roots.
12
9
6
3
2 0
3
6
x
2 4 6 8 10
Copyright © by William H. Sadlier, Inc. All rights reserved.
Find the discriminant of each quadratic equation. Then describe the
number and type of roots of the equation.
1. x2 ⫹ 5x ⫹ 7 ⫽ 0
a 1, b 5, c 7
b2 4ac (5)2 4(1)(7)
25 28 3
no real roots
5. 2x2 ⫺ 19x ⫹ 35 ⫽ 0
a 2, b 19, c 35
(19)2 4(2)(35)
361 280 81
two rational roots
9. 3x2 ⫺ 7x ⫹ 11 ⫽ 0
a 3, b 7, c 11
(7)2 4(3)(11)
49 132 83
no real roots
2. x2 ⫹ 2x ⫹ 6 ⫽ 0
a 1, b 2, c 6
(2)2 4(1)(6)
4 24 20
no real roots
6. 3x2 ⫺ 14x ⫹ 8 ⫽ 0
a 3, b 14, c 8
(14)2 4(3)(8)
196 96 100
two rational roots
10. 4x2 ⫺ 3x ⫹ 5 ⫽ 0
a 4, b 3, c 5
(3)2 4(4)(5)
9 80 71
no real roots
3. x2 ⫺ 5x ⫹ 2 ⫽ 0
a 1, b 5, c 2
(5)2 4(1)(2)
25 8 17
two irrational roots
7. 9x2 ⫺ 6x ⫹ 1 ⫽ 0
a 9, b 6, c 1
(6)2 4(9)(1)
36 36 0
one rational root
11. 4x2 ⫹ 12x ⫹ 9 ⫽ 0
a 4, b 12, c 9
(12)2 4(4)(9)
144 144 0
one rational root
Lesson 10-6, pages 262–263.
4. x2 ⫺ 3x ⫹ 1 ⫽ 0
a 1, b 3, c 1
(3)2 4(1)(1)
945
two irrational roots
8. 25x2 ⫺ 10x ⫹ 1 ⫽ 0
a 25, b 10, c 1
(10)2 4(25)(1)
100 100 0
one rational root
12. 5x2 ⫹ 10x ⫹ 5 ⫽ 0
a 5, b 10, c 5
(10)2 4(5)(5)
100 100 0
one rational root
Chapter 10 257
For More Practice Go To:
Find the discriminant of each quadratic equation. Describe the number
and type of roots of the equation.
a 10, b 11, c 6
(11)2 4(10)(6)
121 (240) 361
two rational roots
16. ⫺5x2 ⫹ 9x ⫹ 3 ⫽ 0
a 5, b 9, c 3
(9)2 4(5)(3)
81 (60) 141
two irrational roots
1
2
1
19. ⫺ 4 x2 ⫺ 3 x ⫺ 2 ⫽ 0
3x2 8x 6 0
a 3, b 8, c 6
64 4(18) 64 72 8
no real roots
5
8
22. 11 x2 ⫺ 3 x ⫺ 7 ⫽ 0
15x2 88x 231 0
a 15, b 88, c 231
7744 4(3465) 21604
two irrational roots
14. 14x2 ⫹ 19x ⫺ 3 ⫽ 0
15. ⫺3x2 ⫹ 7x ⫹ 2 ⫽ 0
a 14, b 19, c 3
(19)2 4(14)(3)
361 (168) 529
two rational roots
17. ⫺16x2 ⫹ 24x ⫺ 9 ⫽ 0
a 16, b 24, c 9
(24)2 4(16)(–9)
576 576 0
one rational root
5
3
6
20. ⫺ 2 x2 ⫺ 4 x ⫺ 5 ⫽ 0
50x2 15x 24 0
a 50, b 15, c 24
225 4(1200) 4575
no real roots
23. ⫺1.4x2 ⫺ 0.5x ⫹ 0.9 ⫽ 0
14x2 5x 9 0
a 14, b 5, c 9
25 4(126) 529
two rational roots
25. For what values of k will the equation
kx2 ⫹ 3x ⫹ 2 ⫽ 0 have no real roots?
Adopt a different point of view.
Solve b2 4ac , 0 for k, where a k, b 3, and c 2:
9
9 4(k)(2) , 0; 9 8k , 0; 9 , 8k; , k; The quadratic
8
equation will have no real roots for all values of k
9
greater than .
8
a 3, b 7, c 2
(7)2 4(3)(2)
49 (24) 73
two irrational roots
18. ⫺121x2 ⫹ 110x ⫺ 25 ⫽ 0
a 121, b 110, c 25
(110)2 4(121)(25)
12,100 12,100 0
one rational root
2
10x2 27x 90 0
a 10, b 27, c 90
729 4(900) 4329
two irrational roots
24. ⫺2.1x2 ⫺ 1.6x ⫹ 0.5 ⫽ 0
21x2 16x 5 0
a 21, b 16, c 5
256 4(105) 676
two rational roots
26. For what values of k will the equation
2x2 ⫺ kx ⫹ 4 ⫽ 0 have 1 real root?
Adopt a different point of view.
Solve b2 4ac 0 for k, where a 2, b k, and
c 4: (k)2 4(2)(4) 0; k2 32 0; k2 32;
k 6 32 6 4 2; The quadratic equation will
have 1 real root when k 4 2 or 4 2 .
27. The graph of a quadratic function goes through the points (⫺5, 2)
and (3, ⫺1). How many solutions does the related equation have?
Explain your answer.
Reason logically: Because the function has points on both sides of the x-axis and
is quadratic, its parabola must intersect the x-axis and in two places. This means
that the related quadratic equation will have two solutions.
258 Chapter 10
3
21. 9 x2 ⫺ 5 x ⫺ 2 ⫽ 0
Copyright © by William H. Sadlier, Inc. All rights reserved.
13. 10x2 ⫹ 11x ⫺ 6 ⫽ 0