Name: ______________________
Class: _________________
Date: _________
ID: A
Using the Discriminant Problem Set
Multiple Choice
Show all of the work that leads to your answer. i.e. b^2 - 4ac = (-3)^2-4(2)(5) = -31
5. The roots of a quadratic equation are real,
rational, and equal when the discriminant is
a. −2
b. 2
c. 0
d. 4
2
1. The roots of the equation 9x + 3x − 4 = 0 are
a. imaginary
b. real, rational, and equal
c. real, rational, and unequal
d. real, irrational, and unequal
2
2. The roots of the equation x − 10x + 25 = 0 are
a. imaginary
b. real and irrational
c. real, rational, and equal
d. real, rational, and unequal
2
6. The equation 2x + 8x + n = 0 has imaginary
roots when n is equal to
a. 10
b. 8
c. 6
d. 4
3. The discriminant of a quadratic equation is 24.
The roots are
a. imaginary
b. real, rational, and equal
c. real, rational, and unequal
d. real, irrational, and unequal
7. For which positive value of m will the equation
2
4x + mx + 9 = 0 have roots that are real, equal,
and rational?
a. 12
b. 9
c. 3
d. 4
2
4. The roots of the equation 2x + 4 = 9x are
a. real, rational, and equal
b. real, rational, and unequal
c. real, irrational, and unequal
d. imaginary
2
8. The roots of the equation 2x − 5 = 0 are
a. imaginary
b. real, rational, and equal
c. real, rational, and unequal
d. real and irrational
1
Name: ______________________
ID: A
11. For which value of k will the roots of
2
2x + kx + 1 = 0 be real?
a. 1
b. 2
c. 3
d. 0
2
9. The roots of the equation 5x − 2x + 1 = 0 are
a. real, rational, and unequal
b. real, rational, and equal
c. real, irrational, and unequal
d. imaginary
10. Which equation has imaginary roots?
2
a. x − 2x + 1 = 0
2
b. x − 2x − 1 = 0
2
c. x − 2x + 5 = 0
2
d. x − 2x − 5 = 0
Short Answer
12. Use the discriminant to determine all values of k
2
that would result in the equation x − kx + 4 = 0
having equal roots.
16. Find the value of k if the roots of the equation
2
x − 6x + k = 0 are equal.
2
17. Solve 2x − 12x + 4 = 0 by completing the square,
expressing the result in simplest radical form.
13. Find all values of k such that the equation
2
3x − 2x + k = 0 has imaginary roots.
-
14. Given the function y = f(x), such that the entire
graph of the function lies above the x-axis.
Explain why the equation f(x) = 0 has no real
solutions.
2
18. Solve the equation 6x − 2x − 3 = 0 and express
the answer in simplest radical form.
15. For what value of k are the roots of
2
2x − 8x + k = 0 equal?
2
ID: A
Using the Discriminant Problem Set
Answer Section
MULTIPLE CHOICE
1. ANS: D
2
2
b − 4ac = 3 − 4(9)(−4) = 9 + 144 = 153
PTS: 2
REF: 081016a2
STA: A2.A.2
KEY: determine nature of roots given equation
2. ANS: C
2
2
b − 4ac = (−10) − 4(1)(25) = 100 − 100 = 0
TOP: Using the Discriminant
PTS: 2
REF: 011102a2
STA: A2.A.2
TOP: Using the Discriminant
KEY: determine nature of roots given equation
3. ANS: D
PTS: 2
REF: 011323a2
STA: A2.A.2
TOP: Using the Discriminant
KEY: determine nature of roots given equation
4. ANS: B
2
2
b − 4ac = (−9) − 4(2)(4) = 81 − 32 = 49
PTS:
KEY:
5. ANS:
TOP:
6. ANS:
2
REF: 011411a2
STA: A2.A.2
TOP: Using the Discriminant
determine nature of roots given equation
C
PTS: 2
REF: 010201b
STA: A2.A.2
Using the Discriminant
KEY: determine equation given nature of roots
A
PTS: 2
REF: 080411b
STA: A2.A.2
KEY: determine equation given nature of roots
7. ANS: A
TOP: Using the Discriminant
PTS: 2
REF: 080516b
STA: A2.A.2
KEY: determine equation given nature of roots
TOP: Using the Discriminant
1
ID: A
8. ANS: D
PTS: 2
REF: 010614b
STA: A2.A.2
KEY: determine nature of roots given equation
9. ANS: D
PTS:
KEY:
10. ANS:
TOP:
11. ANS:
TOP:
TOP: Using the Discriminant
2
REF: 080814b
STA: A2.A.2
TOP: Using the Discriminant
determine nature of roots given equation
C
PTS: 2
REF: 068833siii
STA: A2.A.2
Using the Discriminant
KEY: identify equation given nature of roots
C
PTS: 2
REF: 019031siii
STA: A2.A.2
Using the Discriminant
KEY: determine equation given nature of roots
SHORT ANSWER
12. ANS:
2
b − 4ac = 0
2
k − 4(1)(4) = 0
2
k − 16 = 0
(k + 4)(k − 4) = 0
k = ±4
PTS: 2
REF: 061028a2
STA: A2.A.2
KEY: determine equation given nature of roots
13. ANS:
k>
TOP: Using the Discriminant
1
.
3
PTS: 2
REF: 060423b
STA: A2.A.2
TOP: Using the Discriminant
KEY: determine equation given nature of roots
14. ANS:
Since the graph lies entirely above the x-axis, there is no point on the graph where y = 0.
PTS: 2
KEY: graph
REF: 080525b
STA: A2.A.2
2
TOP: Using the Discriminant
ID: A
15. ANS:
8
PTS: 2
REF: 018418siii
STA: A2.A.2
KEY: determine equation given nature of roots
16. ANS:
9
TOP: Using the Discriminant
PTS: 2
REF: 068512siii
STA: A2.A.2
KEY: determine equation given nature of roots
17. ANS:
2
3 ± 7 . 2x − 12x + 4 = 0
TOP: Using the Discriminant
2
x − 6x + 2 = 0
2
x − 6x = −2
2
x − 6x + 9 = −2 + 9
2
(x − 3) = 7
x−3 = ±
7
x=3±
PTS: 4
18. ANS:
2±
REF: fall0936a2
2
(−2) − 4(6)(−3)
2(6)
PTS: 2
7
=
2±
76
12
=
REF: 011332a2
STA: A2.A.24
2±
4
12
19
=
2 ± 2 19
1 ± 19
=
12
6
STA: A2.A.25
3
TOP: Completing the Square
TOP: Quadratics with Irrational Solutions