What is the product of the roots of 5x^2 − 26x − 25 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = -25/5 = -5.
13
Does the parabola y = 4x^2 + 2x − 14 open upward or downward?
Medium
ExplanationSince a = 4 is positive, the parabola opens upward.
14
What is the sum of the roots of x^2 + 37x + 19 = 0?
Medium
ExplanationBy Vieta's formulas, the sum of the roots equals −b/a = −(37)/1 = -37.
15
What are the solutions to 6x^2 + 66x + 144 = 0?
Medium
ExplanationUsing the quadratic formula with a=6, b=66, c=144, the discriminant is a perfect square, giving x = -8 or x = -3.
16
A rectangle has length (x + 15) and width (x + 6). If the area of the rectangle is 322 square units, what is the value of x (where x is a positive number)?
Medium
Explanation(x + 15)(x + 6) = 322. Testing x = 8: (23)(14) = 322, which checks out (the other algebraic root is negative and not valid here).
17
What is the maximum value of y = -4x^2 − 16x + 24?
Medium
ExplanationSince a = -4 is negative, the parabola opens downward and has a maximum at its vertex, where y = 40.
18
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.
19
What is the nature of the roots of 5x^2 − 5x − 8 = 0?
Medium
ExplanationThe discriminant is b^2 − 4ac = (-5)^2 − 4(5)(-8) = 185, which is positive, so the equation has two distinct real roots.
20
What is the y-intercept of y = 3x^2 − 26x + 16?
Medium
ExplanationThe y-intercept occurs at x = 0: y = 3(0)^2 + (-26)(0) + 16 = 16.
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21
What is the product of the roots of 5x^2 − 26x + 10 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = 10/5 = 2.
22
What is the nature of the roots of 3x^2 − 19x + 3 = 0?
Medium
ExplanationThe discriminant is b^2 − 4ac = (-19)^2 − 4(3)(3) = 325, which is positive, so the equation has two distinct real roots.
ExplanationSum of roots = 4, product of roots = 3, giving x^2 − (4)x + (3) = 0, i.e. x^2 − 4x + 3 = 0.
27
Does the parabola y = -4x^2 + 17x − 25 open upward or downward?
Medium
ExplanationSince a = -4 is negative, the parabola opens downward.
28
What is the product of the roots of x^2 + 40x + 20 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = 20/1 = 20.
29
What is the vertex of the parabola y = 6x^2 − 156x − 3?
Medium
ExplanationThe vertex x-coordinate is −b/(2a) = −(-156)/(2×6) = 13. Substituting gives y = -1017, so the vertex is (13, -1017).
30
What is the maximum value of y = -6x^2 + 84x − 29?
Medium
ExplanationSince a = -6 is negative, the parabola opens downward and has a maximum at its vertex, where y = 265.
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31
What is the product of the roots of 3x^2 − 12x − 42 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = -42/3 = -14.
32
Factor: 66x^2 + 110x
Medium
ExplanationThe GCF of 66 and 110 (besides x) is 11, and factoring it out (along with one x) gives 11x(6x + 10).
33
Is x^2 + 30x + 221 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 = 221, so it is not a perfect square trinomial.
34
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).
35
What is the y-intercept of y = -9x^2 + 7x − 8?
Medium
ExplanationThe y-intercept occurs at x = 0: y = -9(0)^2 + (7)(0) − 8 = -8.
36
What is the nature of the roots of 5x^2 − 6x − 3 = 0?
Medium
ExplanationThe discriminant is b^2 − 4ac = (-6)^2 − 4(5)(-3) = 96, which is positive, so the equation has two distinct real roots.
37
Does the parabola y = 9x^2 + 12x − 8 open upward or downward?
Medium
ExplanationSince a = 9 is positive, the parabola opens upward.
38
What is the y-intercept of y = 4x^2 + 15x − 50?
Medium
ExplanationThe y-intercept occurs at x = 0: y = 4(0)^2 + (15)(0) − 50 = -50.
39
What are the solutions to 4x^2 − 68x + 280 = 0?
Medium
ExplanationUsing the quadratic formula with a=4, b=-68, c=280, the discriminant is a perfect square, giving x = 7 or x = 10.
40
What are the solutions to 5x^2 − 60x + 55 = 0?
Medium
ExplanationUsing the quadratic formula with a=5, b=-60, c=55, the discriminant is a perfect square, giving x = 1 or x = 11.
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41
Complete the square: x^2 − 22x + 137 = (x + h)^2 + k. What are the values of h and k?
ExplanationUsing the quadratic formula with a=2, b=-32, c=120, the discriminant is a perfect square, giving x = 6 or x = 10.
47
What is the nature of the roots of 6x^2 + 10x + 6 = 0?
Medium
ExplanationThe discriminant is b^2 − 4ac = (10)^2 − 4(6)(6) = -44, which is negative, so the equation has no real roots.
48
What is the minimum value of y = 3x^2 + 12x − 24?
Medium
ExplanationSince a = 3 is positive, the parabola opens upward and has a minimum at its vertex, where y = -36.
49
What is the minimum value of y = 3x^2 + 48x + 10?
Medium
ExplanationSince a = 3 is positive, the parabola opens upward and has a minimum at its vertex, where y = -182.
50
What are the solutions to x^2 + 9x − 70 = 0?
Medium
Explanationx^2 + 9x − 70 = (x + 14)(x - 5), so x = -14 or x = 5.
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51
Factor: x^2 − 8x − 84
Medium
Explanationx^2 − 8x − 84 factors as the product of (x − r) for each root: (x - 14)(x + 6).
52
What is the vertex of the parabola y = 4x^2 + 72x + 20?
Medium
ExplanationThe vertex x-coordinate is −b/(2a) = −(72)/(2×4) = -9. Substituting gives y = -304, so the vertex is (-9, -304).
53
What are the solutions to 3x^2 + 15x + 18 = 0?
Medium
ExplanationUsing the quadratic formula with a=3, b=15, c=18, the discriminant is a perfect square, giving x = -3 or x = -2.
54
Solve the inequality: x^2 + 23x + 126 > 0
Medium
Explanationx^2 + 23x + 126 = (x + 14)(x + 9), with roots -14 and -9. Since the parabola opens upward, the expression is positive outside the roots, giving x < -14 or x > -9.
55
What is the y-intercept of y = 8x^2 + 12x + 4?
Medium
ExplanationThe y-intercept occurs at x = 0: y = 8(0)^2 + (12)(0) + 4 = 4.
56
What is the maximum value of y = -3x^2 − 72x + 4?
Medium
ExplanationSince a = -3 is negative, the parabola opens downward and has a maximum at its vertex, where y = 436.
57
What are the solutions to x^2 − 5x + 6 = 0?
Medium
Explanationx^2 − 5x + 6 = (x - 2)(x - 3), so x = 2 or x = 3.
58
Factor: x^2 + 12x + 20
Medium
Explanationx^2 + 12x + 20 factors as the product of (x − r) for each root: (x + 10)(x + 2).
59
Is x^2 + 4x + 12 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 = (2)^2 = 4, which does not match c = 12, so it is not a perfect square trinomial.
Explanationx^2 − 5x = (x - 0)(x - 5), with roots 0 and 5. Since the parabola opens upward, the expression is positive outside the roots, giving x < 0 or x > 5.
What is the nature of the roots of 5x^2 + 17x + 13 = 0?
Medium
ExplanationThe discriminant is b^2 − 4ac = (17)^2 − 4(5)(13) = 29, which is positive, so the equation has two distinct real roots.
66
Which equation has roots -3 and 13?
Medium
ExplanationSum of roots = 10, product of roots = -39, giving x^2 − (10)x + (-39) = 0, i.e. x^2 − 10x − 39 = 0.
67
Solve the inequality: x^2 − 35x + 294 < 0
Medium
Explanationx^2 − 35x + 294 = (x - 14)(x - 21), with roots 14 and 21. Since the parabola opens upward, the expression is negative between the roots, giving 14 < x < 21.
ExplanationThe GCF of 48 and 32 (besides x) is 8, and factoring it out (along with one x) gives 8x(6x + 4).
70
A rectangle has length (x + 11) and width (x + 9). If the area of the rectangle is 399 square units, what is the value of x (where x is a positive number)?
Medium
Explanation(x + 11)(x + 9) = 399. Testing x = 10: (21)(19) = 399, which checks out (the other algebraic root is negative and not valid here).
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71
What is the maximum value of y = -3x^2 − 48x − 24?
Medium
ExplanationSince a = -3 is negative, the parabola opens downward and has a maximum at its vertex, where y = 168.
72
What is the y-intercept of y = 2x^2 + 10x − 27?
Medium
ExplanationThe y-intercept occurs at x = 0: y = 2(0)^2 + (10)(0) − 27 = -27.
73
Factor: x^2 + 12x + 32
Medium
Explanationx^2 + 12x + 32 factors as the product of (x − r) for each root: (x + 8)(x + 4).
74
Solve the inequality: x^2 − 44x + 420 < 0
Medium
Explanationx^2 − 44x + 420 = (x - 14)(x - 30), with roots 14 and 30. Since the parabola opens upward, the expression is negative between the roots, giving 14 < x < 30.
75
What is the axis of symmetry of the parabola y = 6x^2 − 132x − 18?
Medium
ExplanationThe axis of symmetry is x = −b/(2a) = −(-132)/(2×6) = 11.
76
What is the product of the roots of 6x^2 + 34x − 36 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = -36/6 = -6.
77
Complete the square: x^2 + 26x + 140 = (x + h)^2 + k. What are the values of h and k?
ExplanationA trinomial x^2 + bx + c is a perfect square exactly when c = (b/2)^2. Here (b/2)^2 = (1)^2 = 1, which does not match c = 13, so it is not a perfect square trinomial.
79
What are the solutions to x^2 + 21x + 104 = 0?
Medium
Explanationx^2 + 21x + 104 = (x + 8)(x + 13), so x = -13 or x = -8.
80
Does the parabola y = -2x^2 + 6x − 20 open upward or downward?
Medium
ExplanationSince a = -2 is negative, the parabola opens downward.
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81
What is the nature of the roots of 3x^2 + 14x + 12 = 0?
Medium
ExplanationThe discriminant is b^2 − 4ac = (14)^2 − 4(3)(12) = 52, which is positive, so the equation has two distinct real roots.
82
A rectangle has length (x + 14) and width (x + 10). If the area of the rectangle is 1152 square units, what is the value of x (where x is a positive number)?
Medium
Explanation(x + 14)(x + 10) = 1152. Testing x = 22: (36)(32) = 1152, which checks out (the other algebraic root is negative and not valid here).
83
What are the solutions to x^2 + 3x − 88 = 0?
Medium
Explanationx^2 + 3x − 88 = (x + 11)(x - 8), so x = -11 or x = 8.
What is the vertex of the parabola y = -6x^2 + 24x − 6?
Medium
ExplanationThe vertex x-coordinate is −b/(2a) = −(24)/(2×-6) = 2. Substituting gives y = 18, so the vertex is (2, 18).
90
A rectangle has length (x + 14) and width (x + 13). If the area of the rectangle is 342 square units, what is the value of x (where x is a positive number)?
Medium
Explanation(x + 14)(x + 13) = 342. Testing x = 5: (19)(18) = 342, which checks out (the other algebraic root is negative and not valid here).
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91
What is the minimum value of y = 6x^2 − 156x − 26?
Medium
ExplanationSince a = 6 is positive, the parabola opens upward and has a minimum at its vertex, where y = -1040.
92
What is the y-intercept of y = 7x^2 − 22x − 32?
Medium
ExplanationThe y-intercept occurs at x = 0: y = 7(0)^2 + (-22)(0) − 32 = -32.
93
What are the solutions to x^2 + 5x − 6 = 0?
Medium
Explanationx^2 + 5x − 6 = (x + 6)(x - 1), so x = -6 or x = 1.
94
What is the axis of symmetry of the parabola y = 3x^2 − 36x − 9?
Medium
ExplanationThe axis of symmetry is x = −b/(2a) = −(-36)/(2×3) = 6.
95
What is the product of the roots of 8x^2 − 12x − 88 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = -88/8 = -11.
96
What are the solutions to x^2 + 8x − 33 = 0?
Medium
Explanationx^2 + 8x − 33 = (x - 3)(x + 11), so x = -11 or x = 3.
97
What is the minimum value of y = 2x^2 + 60x − 14?
Medium
ExplanationSince a = 2 is positive, the parabola opens upward and has a minimum at its vertex, where y = -464.
98
Does the parabola y = 7x^2 + 14x + 15 open upward or downward?
Medium
ExplanationSince a = 7 is positive, the parabola opens upward.
99
What is the axis of symmetry of the parabola y = -6x^2 + 96x − 6?
Medium
ExplanationThe axis of symmetry is x = −b/(2a) = −(96)/(2×-6) = 8.
100
What is the y-intercept of y = -6x^2 − 25x − 26?
Medium
ExplanationThe y-intercept occurs at x = 0: y = -6(0)^2 + (-25)(0) − 26 = -26.