ECAT Mathematics
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ECAT Mathematics
Chapter 4: Quadratic Equations
100 Questions
90 Minutes
Pass: 60%+
ECAT Prep
0
The discriminant of ax^2+bx+c=0 is:
A
b^2-4ac
B
b^2+4ac
C
-b^2+4ac
D
4ac-b^2
1
If D > 0, the roots of quadratic equation are:
A
Equal and real
B
Unequal and real
C
Imaginary
D
Rational
2
If D = 0, the roots are:
A
Unequal
B
Imaginary
C
Equal and real
D
Irrational
3
If D < 0, the roots are:
A
Real and distinct
B
Equal and real
C
Complex (imaginary)
D
Rational
4
The quadratic formula gives x =
A
(-b ± √D)/2a
B
(b ± √D)/2a
C
(-b ± √D)/a
D
(-b ± D)/2a
5
For ax^2+bx+c=0, sum of roots α+β =
A
b/a
B
-b/a
C
c/a
D
-c/a
6
For ax^2+bx+c=0, product of roots αβ =
A
b/a
B
-b/a
C
c/a
D
-c/a
7
Which equation has roots 2 and 3?
A
x^2-5x+6=0
B
x^2+5x+6=0
C
x^2-5x-6=0
D
x^2+5x-6=0
8
The equation x^2+4x+4=0 has roots that are:
A
Equal (both -2)
B
Unequal real
C
Imaginary
D
Positive
9
If one root of x^2-5x+k=0 is 2, then k =
A
6
B
3
C
10
D
4
10
The roots of x^2+1=0 are:
A
1 and -1
B
i and -i
C
1 and i
D
0 and 1
11
A quadratic equation always has exactly:
A
One root
B
Two roots (counting multiplicity, in C)
C
Three roots
D
No roots
12
For x^2-6x+9=0, the roots are:
A
3 and -3
B
3 and 3 (double root)
C
6 and 9
D
-3 and -3
13
The nature of roots depends on:
A
Only the coefficients a and b
B
The discriminant
C
The sum of roots
D
The product of roots
14
If roots of x^2+px+q=0 are equal, then p^2 =
A
q
B
4q
C
2q
D
q^2
15
The quadratic equation with roots -1 and 4 is:
A
x^2-3x-4=0
B
x^2+3x-4=0
C
x^2-5x+4=0
D
x^2+5x-4=0
16
For 2x^2-3x+1=0, sum of roots =
A
3/2
B
-3/2
C
1/2
D
-1/2
17
For 2x^2-3x+1=0, product of roots =
A
3/2
B
1/2
C
-1/2
D
2
18
Which quadratic has irrational roots?
A
x^2-4=0
B
x^2-2=0
C
x^2+1=0
D
x^2-9=0
19
The roots of 3x^2+7x+2=0 are:
A
-1/3 and -2
B
-2 and 2
C
1/3 and 2
D
-1 and -2
20
If α and β are roots, α^2+β^2 =
A
(α+β)^2 + 2αβ
B
(α+β)^2 - 2αβ
C
αβ^2 + α^2β
D
(α-β)^2
21
If α and β are roots, (α-β)^2 =
A
(α+β)^2-4αβ
B
(α+β)^2+4αβ
C
4αβ-(α+β)^2
D
αβ
22
The vertex of parabola y=ax^2+bx+c is at:
A
(-b/2a, f(-b/2a))
B
(b/2a, f(b/2a))
C
(-b/a, 0)
D
(0, c)
23
The equation x^2+bx+c=0 has both roots negative if:
A
b>0 and c>0
B
b<0 and c>0
C
b>0 and c<0
D
b<0 and c<0
24
Solving x^2-7x+12=0 by factoring gives:
A
(x-3)(x-4)=0
B
(x+3)(x+4)=0
C
(x-3)(x+4)=0
D
(x+3)(x-4)=0
25
The equation that cannot be solved by factoring (over integers) for x^2-3x+1=0 because:
A
D=5, irrational roots
B
D=0
C
D<0
D
a=0
26
The maximum or minimum value of y=ax^2+bx+c occurs at x=
A
b/2a
B
-b/2a
C
a/2b
D
-a/2b
27
For y=ax^2+bx+c, if a>0, the parabola opens:
A
Downward
B
Upward
C
Sideways
D
Neither
28
For y=ax^2+bx+c, if a<0, the parabola opens:
A
Upward
B
Downward
C
Sideways
D
Horizontally
29
An equation reducible to quadratic: x^4-5x^2+4=0 can be solved using substitution:
A
u = x^4
B
u = x^2
C
u = x^3
D
u = 1/x
30
Solving u^2-5u+4=0: u =
A
4 or 1
B
5 or 0
C
2 or 2
D
-4 or -1
31
The reciprocal equation ax^2+bx+a=0 has roots that are:
A
Equal
B
Reciprocals of each other
C
Conjugates
D
Negative of each other
32
The discriminant of x^2+2x+5=0 is:
A
-16
B
16
C
-24
D
0
33
The roots of x^2+2x+5=0 are:
A
-1±2i
B
1±2i
C
-1±√6
D
1±√6
34
Completing the square for x^2+6x+2=0 gives:
A
(x+3)^2=7
B
(x+3)^2=-7
C
(x+3)^2=11
D
(x-3)^2=7
35
If α and β are roots of 3x^2-5x+2=0, then 1/α+1/β =
A
5/2
B
2/5
C
5/3
D
3/5
36
Quadratic equation with roots (1+√2) and (1-√2) is:
A
x^2-2x-1=0
B
x^2+2x+1=0
C
x^2-2x+1=0
D
x^2+2x-1=0
37
The equation 4x^2-4x+1=0 factors as:
A
(2x-1)^2=0
B
(2x+1)^2=0
C
(4x-1)(x-1)=0
D
(2x-1)(2x+1)=0
38
For roots α,β of ax^2+bx+c=0, α^3+β^3 =
A
(α+β)^3-3αβ(α+β)
B
(α+β)^3+3αβ
C
(α+β)^3-3αβ
D
αβ(α+β)
39
The number of real roots of x^2+x+1=0 is:
A
2
B
1
C
0
D
Infinite
40
If one root is double the other for kx^2+3x+2=0, then k =
A
9/8
B
8/9
C
3/2
D
2/3
41
The equation with one root at 0 is:
A
x^2+3x+2=0
B
x^2+3x=0
C
x^2+1=0
D
x^2-5x+6=0
42
If both roots of x^2+px+q=0 are positive, then:
A
p>0 and q>0
B
p<0 and q>0
C
p>0 and q<0
D
p<0 and q<0
43
The quadratic x^2-√5x+1=0 has roots that are:
A
Rational
B
Equal
C
Irrational and real
D
Complex
44
Vieta's formulas relate roots to:
A
Coefficients of the quadratic
B
The discriminant only
C
The vertex only
D
The axis of symmetry
45
The axis of symmetry of y=2x^2-8x+3 is:
A
x=2
B
x=-2
C
x=4
D
x=1
46
If α-β=1 and αβ=6, then α+β =
A
5
B
√25=5
C
√(1+24)=5
D
Both b and c
47
For x^2-5x+6=0, the roots are:
A
2 and 3
B
-2 and -3
C
1 and 6
D
2 and -3
48
If x=1 is a root of x^2+kx-2=0, then k =
A
1
B
-1
C
2
D
-2
49
The quadratic with sum of roots 4 and product of roots -5 is:
A
x^2-4x-5=0
B
x^2+4x-5=0
C
x^2-4x+5=0
D
x^2+4x+5=0
50
The equation x^2=9 has roots:
A
3 only
B
-3 only
C
3 and -3
D
9 and -9
51
If a=1,b=-2,c=-3, D equals:
A
16
B
-16
C
8
D
4
52
The graph of y=x^2-4 crosses x-axis at:
A
x=4 and x=-4
B
x=2 and x=-2
C
x=0
D
x=4 only
53
When will x^2+bx+c=0 have one positive and one negative root?
A
c>0
B
c<0
C
b>0
D
b<0
54
Which of the following is an identity (true for all x)?
A
x^2+x=0
B
x^2-1=(x-1)(x+1)
C
x^2=x
D
x^2=4
55
x^2+6x+k=0 has equal roots when k=
A
9
B
3
C
12
D
6
56
The standard form of quadratic equation is:
A
ax+b=0
B
ax^2+bx+c=0 (a≠0)
C
ax^3+bx^2+c=0
D
a/x^2=0
57
For 5x^2-13x+6=0, product of roots is:
A
6/5
B
-6/5
C
13/5
D
-13/5
58
The roots of (x-2)(x+3)=0 are:
A
2 and 3
B
-2 and 3
C
2 and -3
D
-2 and -3
59
For equation x^2-3x+p=0 if roots are real and unequal:
A
p > 9/4
B
p < 9/4
C
p = 9/4
D
p > 3
60
If α and β are roots, then α^2β+αβ^2 =
A
αβ(α+β)
B
α+β
C
αβ
D
(α+β)^2
61
For x^2-4x+4=0, what type are the roots?
A
Imaginary
B
Irrational
C
Rational and equal
D
Irrational and equal
62
If roots of x^2+px+1=0 are real, then p lies in:
A
p>2
B
p<-2
C
p≤-2 or p≥2
D
p=±2
63
The quadratic x^2-x-6=0 factored gives:
A
(x-3)(x+2)=0
B
(x+3)(x-2)=0
C
(x-6)(x+1)=0
D
(x-3)(x-2)=0
64
If α and β are roots, (α+1)(β+1) =
A
αβ+α+β+1
B
αβ+1
C
α+β+1
D
αβ-α-β+1
65
The condition for one root to be reciprocal of the other is:
A
a=b
B
a=c
C
a=-c
D
b=0
66
The equation 4x^2-12x+9=0 has roots:
A
3/2 and 3/2
B
3/2 and -3/2
C
3 and 1/3
D
2 and 3
67
A quadratic equation is degree:
A
1
B
2
C
3
D
4
68
The maximum number of roots of a quadratic equation is:
A
1
B
2
C
3
D
4
69
If both roots of x^2+px+q=0 are equal in magnitude but opposite in sign:
A
p=0
B
q=0
C
p=q
D
p=-q
70
The graph of y=x^2-6x+8 has minimum value:
A
-1
B
1
C
0
D
-8
71
The equation with roots 1/2 and -3 is:
A
2x^2+5x-3=0
B
2x^2-5x-3=0
C
2x^2+5x+3=0
D
x^2+5x-3=0
72
If 3+i is one root of a quadratic with real coefficients, the other is:
A
3+i
B
3-i
C
-3+i
D
-3-i
73
The number of real roots of x^2+2x+2=0:
A
2
B
1
C
0
D
Infinite
74
For kx^2+2x+1=0 to have two equal roots, k=
A
1
B
0
C
2
D
-1
75
The sum of squares of roots α^2+β^2 for x^2-3x+2=0 is:
A
5
B
9
C
4
D
7
76
Which method is most general for solving quadratic equations?
A
Factoring
B
Completing the square
C
Quadratic formula
D
Inspection
77
The quadratic 2x^2+5x-3=0 has discriminant:
A
25+24=49
B
25-24=1
C
5+6=11
D
25+12=37
78
Roots of 2x^2+5x-3=0:
A
-3 and 1/2
B
3 and -1/2
C
-3 and -1/2
D
3 and 1/2
79
If the roots of 3x^2+px+12=0 are equal, then p =
A
±12
B
±13
C
12
D
±6
80
For x^2+3x-10=0, the roots are:
A
2 and -5
B
-2 and 5
C
5 and 2
D
-5 and -2
81
The product of roots of (x-3)(x+5)=0 is:
A
-15
B
15
C
2
D
-2
82
For a quadratic y=x^2-2x+1, the vertex is at:
A
(1,0)
B
(0,1)
C
(-1,0)
D
(2,1)
83
The solution set of x^2-4≥0 is:
A
(-2,2)
B
(-∞,-2]∪[2,∞)
C
[-2,2]
D
(2,∞)
84
The solution set of x^2-4<0 is:
A
(-∞,-2)∪(2,∞)
B
(-2,2)
C
[-2,2]
D
(2,∞)
85
For the equation x^2-kx+k-1=0, one root is always:
A
0
B
1
C
-1
D
2
86
If α,β are roots of x^2-5x+6=0, then α/β + β/α =
A
25/6-2
B
(α^2+β^2)/αβ
C
13/6
D
Both b and c
87
A quadratic equation has exactly 2 roots (in complex numbers):
A
Never
B
Only if a≠0
C
Only if D>0
D
Only for perfect squares
88
The equation px^2+(p-q)x-q=0 has roots:
A
p and q
B
-1 and q/p
C
-1 and 1
D
-1 and q/p
89
What is a 'pure quadratic'?
A
ax^2+c=0 (no x term)
B
ax^2+bx=0
C
ax^2+bx+c with b≠0
D
Linear equation
90
The roots of 4x^2-8x+3=0 are:
A
1/2 and 3/2
B
-1/2 and 3/2
C
1/2 and -3/2
D
3 and 4
91
Completing the square of x^2+8x gives:
A
(x+4)^2-16
B
(x+4)^2+16
C
(x+8)^2
D
(x+8)^2-64
92
The condition for roots to be rational is:
A
D=0
B
D is a perfect square (≥0)
C
D<0
D
D is a perfect cube
93
For roots in the ratio 2:3 of x^2+5x+k=0:
A
k=6
B
k=3
C
k=4
D
k=8
94
The equation (x-1)^2 = 4 has roots:
A
3 and -1
B
-3 and 1
C
5 and -3
D
1 and -1
95
The sum of roots of 7x^2-3x+1=0 is:
A
3/7
B
-3/7
C
1/7
D
7/3
96
The product of roots of 7x^2-3x+1=0 is:
A
1/7
B
3/7
C
-1/7
D
7
97
If one root of x^2-(k+1)x+k+4=0 is -1, then k =
A
-2
B
2
C
-3
D
3
98
A quadratic equation can have at most how many real roots?
A
1
B
2
C
3
D
Infinite
99
The graph of y=-x^2+4x-3 has maximum value:
A
1
B
4
C
3
D
-3
0
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