ExplanationThe GCF of 48 and -40 (besides x) is 8, and factoring it out (along with one x) gives 8x(6x − 5).
2
What is the vertex of the parabola y = x^2 + 24x + 13?
Medium
ExplanationThe vertex x-coordinate is −b/(2a) = −(24)/(2×1) = -12. Substituting gives y = -131, so the vertex is (-12, -131).
3
Factor: x^2 + 2x − 195
Medium
Explanationx^2 + 2x − 195 factors as the product of (x − r) for each root: (x - 13)(x + 15).
4
Is x^2 − 30x + 213 a perfect square trinomial?
Medium
ExplanationA trinomial x^2 + bx + c is a perfect square exactly when c = (b/2)^2. Here (b/2)^2 = (-15)^2 = 225, which does not match c = 213, so it is not a perfect square trinomial.
5
Is x^2 − 28x + 198 a perfect square trinomial?
Medium
ExplanationA trinomial x^2 + bx + c is a perfect square exactly when c = (b/2)^2. Here (b/2)^2 = (-14)^2 = 196, which does not match c = 198, so it is not a perfect square trinomial.
6
Which equation has roots -5 and 2?
Medium
ExplanationSum of roots = -3, product of roots = -10, giving x^2 − (-3)x + (-10) = 0, i.e. x^2 + 3x − 10 = 0.
7
What is the nature of the roots of 4x^2 + 20x + 6 = 0?
Medium
ExplanationThe discriminant is b^2 − 4ac = (20)^2 − 4(4)(6) = 304, which is positive, so the equation has two distinct real roots.
8
What are the solutions to 6x^2 − 12x − 18 = 0?
Medium
ExplanationUsing the quadratic formula with a=6, b=-12, c=-18, the discriminant is a perfect square, giving x = -1 or x = 3.
9
Factor: 10x^2 − 50x
Medium
ExplanationThe GCF of 10 and -50 (besides x) is 5, and factoring it out (along with one x) gives 5x(2x − 10).
10
Does the parabola y = -9x^2 + 29x − 29 open upward or downward?
Medium
ExplanationSince a = -9 is negative, the parabola opens downward.
What is the sum of the roots of x^2 − 27x + 7 = 0?
Medium
ExplanationBy Vieta's formulas, the sum of the roots equals −b/a = −(-27)/1 = 27.
15
Factor: 55x^2 + 99x
Medium
ExplanationThe GCF of 55 and 99 (besides x) is 11, and factoring it out (along with one x) gives 11x(5x + 9).
16
What is the vertex of the parabola y = 2x^2 + 36x + 19?
Medium
ExplanationThe vertex x-coordinate is −b/(2a) = −(36)/(2×2) = -9. Substituting gives y = -143, so the vertex is (-9, -143).
17
Factor: 88x^2 + 55x
Medium
ExplanationThe GCF of 88 and 55 (besides x) is 11, and factoring it out (along with one x) gives 11x(8x + 5).
18
What is the axis of symmetry of the parabola y = 6x^2 − 24x − 12?
Medium
ExplanationThe axis of symmetry is x = −b/(2a) = −(-24)/(2×6) = 2.
19
Does the parabola y = 6x^2 + 24x + 5 open upward or downward?
Medium
ExplanationSince a = 6 is positive, the parabola opens upward.
20
A rectangle has length (x + 11) and width (x + 4). If the area of the rectangle is 78 square units, what is the value of x (where x is a positive number)?
Medium
Explanation(x + 11)(x + 4) = 78. Testing x = 2: (13)(6) = 78, which checks out (the other algebraic root is negative and not valid here).
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21
Which equation has roots -6 and 2?
Medium
ExplanationSum of roots = -4, product of roots = -12, giving x^2 − (-4)x + (-12) = 0, i.e. x^2 + 4x − 12 = 0.
22
Factor: x^2 + 11x + 10
Medium
Explanationx^2 + 11x + 10 factors as the product of (x − r) for each root: (x + 10)(x + 1).
23
What is the axis of symmetry of the parabola y = -3x^2 − 54x − 22?
Medium
ExplanationThe axis of symmetry is x = −b/(2a) = −(-54)/(2×-3) = -9.
24
What is the maximum value of y = -2x^2 − 28x − 11?
Medium
ExplanationSince a = -2 is negative, the parabola opens downward and has a maximum at its vertex, where y = 87.
25
What are the solutions to x^2 − 12x + 20 = 0?
Medium
Explanationx^2 − 12x + 20 = (x - 2)(x - 10), so x = 2 or x = 10.
26
What are the solutions to x^2 − 27x + 180 = 0?
Medium
Explanationx^2 − 27x + 180 = (x - 12)(x - 15), so x = 12 or x = 15.
27
Factor: x^2 + 13x + 12
Medium
Explanationx^2 + 13x + 12 factors as the product of (x − r) for each root: (x + 1)(x + 12).
28
Factor: 30x^2 + 45x
Medium
ExplanationThe GCF of 30 and 45 (besides x) is 5, and factoring it out (along with one x) gives 5x(6x + 9).
29
What is the vertex of the parabola y = 3x^2 − 4?
Medium
ExplanationThe vertex x-coordinate is −b/(2a) = −(0)/(2×3) = 0. Substituting gives y = -4, so the vertex is (0, -4).
30
What is the product of the roots of 8x^2 + 38x + 24 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = 24/8 = 3.
A rectangle has length (x + 4) and width (x + 12). If the area of the rectangle is 768 square units, what is the value of x (where x is a positive number)?
Medium
Explanation(x + 4)(x + 12) = 768. Testing x = 20: (24)(32) = 768, which checks out (the other algebraic root is negative and not valid here).
37
What are the solutions to 2x^2 + 12x − 14 = 0?
Medium
ExplanationUsing the quadratic formula with a=2, b=12, c=-14, the discriminant is a perfect square, giving x = -7 or x = 1.
38
What is the y-intercept of y = -6x^2 + 12x − 35?
Medium
ExplanationThe y-intercept occurs at x = 0: y = -6(0)^2 + (12)(0) − 35 = -35.
39
Is x^2 − 20x + 100 a perfect square trinomial?
Medium
ExplanationA trinomial x^2 + bx + c is a perfect square exactly when c = (b/2)^2. Here (b/2)^2 = (-10)^2 = 100, which matches c, so it is a perfect square trinomial.
40
Factor: 18x^2 + 4x
Medium
ExplanationThe GCF of 18 and 4 (besides x) is 2, and factoring it out (along with one x) gives 2x(9x + 2).
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41
What are the solutions to x^2 + x − 42 = 0?
Medium
Explanationx^2 + x − 42 = (x + 7)(x - 6), so x = -7 or x = 6.
42
What is the vertex of the parabola y = 4x^2 + 112x + 18?
Medium
ExplanationThe vertex x-coordinate is −b/(2a) = −(112)/(2×4) = -14. Substituting gives y = -766, so the vertex is (-14, -766).
43
What is the minimum value of y = x^2 + 4x + 16?
Medium
ExplanationSince a = 1 is positive, the parabola opens upward and has a minimum at its vertex, where y = 12.
44
Does the parabola y = 6x^2 − 28x + 22 open upward or downward?
Medium
ExplanationSince a = 6 is positive, the parabola opens upward.
45
Complete the square: x^2 − 18x + 111 = (x + h)^2 + k. What are the values of h and k?
ExplanationUsing the quadratic formula with a=3, b=18, c=-81, the discriminant is a perfect square, giving x = -9 or x = 3.
47
What is the y-intercept of y = -7x^2 − 27x − 13?
Medium
ExplanationThe y-intercept occurs at x = 0: y = -7(0)^2 + (-27)(0) − 13 = -13.
48
Solve the inequality: x^2 − 15x − 34 < 0
Medium
Explanationx^2 − 15x − 34 = (x + 2)(x - 17), with roots -2 and 17. Since the parabola opens upward, the expression is negative between the roots, giving -2 < x < 17.
49
What is the vertex of the parabola y = 2x^2 − 40x − 24?
Medium
ExplanationThe vertex x-coordinate is −b/(2a) = −(-40)/(2×2) = 10. Substituting gives y = -224, so the vertex is (10, -224).
50
What are the solutions to 2x^2 − 22x + 20 = 0?
Medium
ExplanationUsing the quadratic formula with a=2, b=-22, c=20, the discriminant is a perfect square, giving x = 1 or x = 10.
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51
Which equation has roots 5 and -12?
Medium
ExplanationSum of roots = -7, product of roots = -60, giving x^2 − (-7)x + (-60) = 0, i.e. x^2 + 7x − 60 = 0.
52
Factor: 20x^2 + 90x
Medium
ExplanationThe GCF of 20 and 90 (besides x) is 10, and factoring it out (along with one x) gives 10x(2x + 9).
What is the product of the roots of x^2 + 11x + 14 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = 14/1 = 14.
55
Is x^2 − 16x + 54 a perfect square trinomial?
Medium
ExplanationA trinomial x^2 + bx + c is a perfect square exactly when c = (b/2)^2. Here (b/2)^2 = (-8)^2 = 64, which does not match c = 54, so it is not a perfect square trinomial.
56
Factor: x^2 − 225
Medium
Explanationx^2 − 225 = x^2 − 15^2 = (x − 15)(x + 15), using the difference-of-squares identity.
57
What is the y-intercept of y = -2x^2 + 11x + 9?
Medium
ExplanationThe y-intercept occurs at x = 0: y = -2(0)^2 + (11)(0) + 9 = 9.
58
Factor: 30x^2 − 15x
Medium
ExplanationThe GCF of 30 and -15 (besides x) is 3, and factoring it out (along with one x) gives 3x(10x − 5).
59
What is the product of the roots of 8x^2 − 4x − 16 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = -16/8 = -2.
60
Does the parabola y = x^2 − 15x − 6 open upward or downward?
Medium
ExplanationSince a = 1 is positive, the parabola opens upward.
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61
What is the vertex of the parabola y = 6x^2 + 108x − 20?
Medium
ExplanationThe vertex x-coordinate is −b/(2a) = −(108)/(2×6) = -9. Substituting gives y = -506, so the vertex is (-9, -506).
62
Factor: x^2 − 7x − 60
Medium
Explanationx^2 − 7x − 60 factors as the product of (x − r) for each root: (x + 5)(x - 12).
63
Complete the square: x^2 − 28x + 214 = (x + h)^2 + k. What are the values of h and k?
ExplanationSum of roots = 10, product of roots = -39, giving x^2 − (10)x + (-39) = 0, i.e. x^2 − 10x − 39 = 0.
65
What are the solutions to x^2 + 17x + 52 = 0?
Medium
Explanationx^2 + 17x + 52 = (x + 13)(x + 4), so x = -13 or x = -4.
66
What is the product of the roots of 5x^2 − 29x + 35 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = 35/5 = 7.
67
Is x^2 + 24x + 144 a perfect square trinomial?
Medium
ExplanationA trinomial x^2 + bx + c is a perfect square exactly when c = (b/2)^2. Here (b/2)^2 = (12)^2 = 144, which matches c, so it is a perfect square trinomial.
68
What are the solutions to 6x^2 + 78x + 132 = 0?
Medium
ExplanationUsing the quadratic formula with a=6, b=78, c=132, the discriminant is a perfect square, giving x = -11 or x = -2.
69
Factor: x^2 − 11x + 28
Medium
Explanationx^2 − 11x + 28 factors as the product of (x − r) for each root: (x - 7)(x - 4).
70
What is the y-intercept of y = -5x^2 − 25x − 27?
Medium
ExplanationThe y-intercept occurs at x = 0: y = -5(0)^2 + (-25)(0) − 27 = -27.
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71
What is the value of -x^2 − 18x − 23 when x = -10?
ExplanationThe y-intercept occurs at x = 0: y = 8(0)^2 + (12)(0) + 4 = 4.
74
What is the y-intercept of y = 5x^2 + 3x + 12?
Medium
ExplanationThe y-intercept occurs at x = 0: y = 5(0)^2 + (3)(0) + 12 = 12.
75
What is the nature of the roots of 2x^2 + 4x − 15 = 0?
Medium
ExplanationThe discriminant is b^2 − 4ac = (4)^2 − 4(2)(-15) = 136, which is positive, so the equation has two distinct real roots.
76
What is the y-intercept of y = 8x^2 + 12?
Medium
ExplanationThe y-intercept occurs at x = 0: y = 8(0)^2 + (0)(0) + 12 = 12.
77
A rectangle has length (x + 1) and width (x + 14). If the area of the rectangle is 1014 square units, what is the value of x (where x is a positive number)?
Medium
Explanation(x + 1)(x + 14) = 1014. Testing x = 25: (26)(39) = 1014, which checks out (the other algebraic root is negative and not valid here).
78
Solve the inequality: x^2 − 35x + 264 < 0
Medium
Explanationx^2 − 35x + 264 = (x - 11)(x - 24), with roots 11 and 24. Since the parabola opens upward, the expression is negative between the roots, giving 11 < x < 24.
79
Solve the inequality: x^2 − 50x + 525 > 0
Medium
Explanationx^2 − 50x + 525 = (x - 15)(x - 35), with roots 15 and 35. Since the parabola opens upward, the expression is positive outside the roots, giving x < 15 or x > 35.
80
What is the axis of symmetry of the parabola y = -2x^2 + 32x + 8?
Medium
ExplanationThe axis of symmetry is x = −b/(2a) = −(32)/(2×-2) = 8.
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81
What are the solutions to 2x^2 − 16x − 18 = 0?
Medium
ExplanationUsing the quadratic formula with a=2, b=-16, c=-18, the discriminant is a perfect square, giving x = -1 or x = 9.
82
What is the nature of the roots of 5x^2 − 7x − 20 = 0?
Medium
ExplanationThe discriminant is b^2 − 4ac = (-7)^2 − 4(5)(-20) = 449, which is positive, so the equation has two distinct real roots.
83
Which equation has roots -11 and 7?
Medium
ExplanationSum of roots = -4, product of roots = -77, giving x^2 − (-4)x + (-77) = 0, i.e. x^2 + 4x − 77 = 0.
84
Does the parabola y = -9x^2 + 15x − 24 open upward or downward?
Medium
ExplanationSince a = -9 is negative, the parabola opens downward.
85
A rectangle has length (x + 5) and width (x + 14). If the area of the rectangle is 532 square units, what is the value of x (where x is a positive number)?
Medium
Explanation(x + 5)(x + 14) = 532. Testing x = 14: (19)(28) = 532, which checks out (the other algebraic root is negative and not valid here).
86
What is the product of the roots of 2x^2 + 19x + 4 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = 4/2 = 2.
87
What is the axis of symmetry of the parabola y = 4x^2 + 48x − 24?
Medium
ExplanationThe axis of symmetry is x = −b/(2a) = −(48)/(2×4) = -6.
88
Factor: x^2 + 18x + 77
Medium
Explanationx^2 + 18x + 77 factors as the product of (x − r) for each root: (x + 7)(x + 11).
89
A rectangle has length (x + 12) and width (x + 8). If the area of the rectangle is 837 square units, what is the value of x (where x is a positive number)?
Medium
Explanation(x + 12)(x + 8) = 837. Testing x = 19: (31)(27) = 837, which checks out (the other algebraic root is negative and not valid here).
90
What is the vertex of the parabola y = 6x^2 − 12x − 15?
Medium
ExplanationThe vertex x-coordinate is −b/(2a) = −(-12)/(2×6) = 1. Substituting gives y = -21, so the vertex is (1, -21).
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91
What is the minimum value of y = 6x^2 + 132x − 9?
Medium
ExplanationSince a = 6 is positive, the parabola opens upward and has a minimum at its vertex, where y = -735.
92
What is the maximum value of y = -5x^2 + 30x + 30?
Medium
ExplanationSince a = -5 is negative, the parabola opens downward and has a maximum at its vertex, where y = 75.
93
What is the y-intercept of y = x^2 + 27x − 46?
Medium
ExplanationThe y-intercept occurs at x = 0: y = 1(0)^2 + (27)(0) − 46 = -46.
94
Factor: x^2 − 6x + 8
Medium
Explanationx^2 − 6x + 8 factors as the product of (x − r) for each root: (x - 4)(x - 2).
95
Is x^2 − 18x + 83 a perfect square trinomial?
Medium
ExplanationA trinomial x^2 + bx + c is a perfect square exactly when c = (b/2)^2. Here (b/2)^2 = (-9)^2 = 81, which does not match c = 83, so it is not a perfect square trinomial.
96
What is the product of the roots of 2x^2 + 7x − 12 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = -12/2 = -6.
97
What is the value of -4x^2 − 20x + 20 when x = -4?