A rectangle has length (x + 3) and width (x + 10). If the area of the rectangle is 980 square units, what is the value of x (where x is a positive number)?
Medium
Explanation(x + 3)(x + 10) = 980. Testing x = 25: (28)(35) = 980, which checks out (the other algebraic root is negative and not valid here).
7
What is the vertex of the parabola y = x^2 − 10x − 15?
Medium
ExplanationThe vertex x-coordinate is −b/(2a) = −(-10)/(2×1) = 5. Substituting gives y = -40, so the vertex is (5, -40).
8
What is the axis of symmetry of the parabola y = -2x^2 + 52x − 5?
Medium
ExplanationThe axis of symmetry is x = −b/(2a) = −(52)/(2×-2) = 13.
9
A rectangle has length (x + 3) and width (x + 5). If the area of the rectangle is 35 square units, what is the value of x (where x is a positive number)?
Medium
Explanation(x + 3)(x + 5) = 35. Testing x = 2: (5)(7) = 35, which checks out (the other algebraic root is negative and not valid here).
10
What is the product of the roots of x^2 + 4x + 7 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = 7/1 = 7.
Explanationx^2 + x − 30 = (x + 6)(x - 5), with roots -6 and 5. Since the parabola opens upward, the expression is negative between the roots, giving -6 < x < 5.
13
Which equation has roots -9 and 11?
Medium
ExplanationSum of roots = 2, product of roots = -99, giving x^2 − (2)x + (-99) = 0, i.e. x^2 − 2x − 99 = 0.
14
Factor: x^2 − 13x + 30
Medium
Explanationx^2 − 13x + 30 factors as the product of (x − r) for each root: (x - 3)(x - 10).
15
Is x^2 + 2x + 10 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 = (1)^2 = 1, which does not match c = 10, so it is not a perfect square trinomial.
16
What is the product of the roots of 3x^2 + 16x − 30 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = -30/3 = -10.
17
What is the product of the roots of 4x^2 + 8x − 28 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = -28/4 = -7.
18
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).
19
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.
20
What is the y-intercept of y = -6x^2 + 16x + 36?
Medium
ExplanationThe y-intercept occurs at x = 0: y = -6(0)^2 + (16)(0) + 36 = 36.
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21
Which equation has roots 1 and -4?
Medium
ExplanationSum of roots = -3, product of roots = -4, giving x^2 − (-3)x + (-4) = 0, i.e. x^2 + 3x − 4 = 0.
22
What is the axis of symmetry of the parabola y = -2x^2 − 52x + 17?
Medium
ExplanationThe axis of symmetry is x = −b/(2a) = −(-52)/(2×-2) = -13.
23
Does the parabola y = -x^2 − 10x − 18 open upward or downward?
Medium
ExplanationSince a = -1 is negative, the parabola opens downward.
24
What is the nature of the roots of 2x^2 − 4x − 2 = 0?
Medium
ExplanationThe discriminant is b^2 − 4ac = (-4)^2 − 4(2)(-2) = 32, which is positive, so the equation has two distinct real roots.
25
What is the vertex of the parabola y = -5x^2 + 50x − 5?
Medium
ExplanationThe vertex x-coordinate is −b/(2a) = −(50)/(2×-5) = 5. Substituting gives y = 120, so the vertex is (5, 120).
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 = 240, so it is not a perfect square trinomial.
28
What is the y-intercept of y = 2x^2 + 4x − 41?
Medium
ExplanationThe y-intercept occurs at x = 0: y = 2(0)^2 + (4)(0) − 41 = -41.
29
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).
Explanationx^2 + 5x − 36 = (x + 9)(x - 4), with roots -9 and 4. Since the parabola opens upward, the expression is negative between the roots, giving -9 < x < 4.
32
What is the y-intercept of y = -5x^2 + 10x − 5?
Medium
ExplanationThe y-intercept occurs at x = 0: y = -5(0)^2 + (10)(0) − 5 = -5.
33
What is the product of the roots of 7x^2 + 30x + 28 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = 28/7 = 4.
34
Complete the square: x^2 − 14x + 66 = (x + h)^2 + k. What are the values of h and k?
What is the vertex of the parabola y = 6x^2 + 180x − 27?
Medium
ExplanationThe vertex x-coordinate is −b/(2a) = −(180)/(2×6) = -15. Substituting gives y = -1377, so the vertex is (-15, -1377).
40
What is the vertex of the parabola y = 2x^2 − 20x − 5?
Medium
ExplanationThe vertex x-coordinate is −b/(2a) = −(-20)/(2×2) = 5. Substituting gives y = -55, so the vertex is (5, -55).
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41
A rectangle has length (x + 7) and width (x + 12). If the area of the rectangle is 644 square units, what is the value of x (where x is a positive number)?
Medium
Explanation(x + 7)(x + 12) = 644. Testing x = 16: (23)(28) = 644, which checks out (the other algebraic root is negative and not valid here).
42
What is the minimum value of y = 6x^2 + 72x + 17?
Medium
ExplanationSince a = 6 is positive, the parabola opens upward and has a minimum at its vertex, where y = -199.
43
What is the product of the roots of 7x^2 − 8x + 112 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = 112/7 = 16.
44
What are the solutions to 2x^2 + 8x − 10 = 0?
Medium
ExplanationUsing the quadratic formula with a=2, b=8, c=-10, the discriminant is a perfect square, giving x = -5 or x = 1.
45
Is x^2 − 16x + 76 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 = 76, so it is not a perfect square trinomial.
A rectangle has length (x + 1) and width (x + 2). If the area of the rectangle is 600 square units, what is the value of x (where x is a positive number)?
Medium
Explanation(x + 1)(x + 2) = 600. Testing x = 23: (24)(25) = 600, which checks out (the other algebraic root is negative and not valid here).
54
What is the y-intercept of y = -4x^2 + 4x + 30?
Medium
ExplanationThe y-intercept occurs at x = 0: y = -4(0)^2 + (4)(0) + 30 = 30.
55
What is the product of the roots of 5x^2 − 30x − 25 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = -25/5 = -5.
56
Does the parabola y = 4x^2 + 9x − 23 open upward or downward?
Medium
ExplanationSince a = 4 is positive, the parabola opens upward.
57
What is the product of the roots of 3x^2 − 9x + 21 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = 21/3 = 7.
58
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.
What is the nature of the roots of 5x^2 + 9x + 14 = 0?
Medium
ExplanationThe discriminant is b^2 − 4ac = (9)^2 − 4(5)(14) = -199, which is negative, so the equation has no real roots.
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61
What is the axis of symmetry of the parabola y = 5x^2 + 100x − 8?
Medium
ExplanationThe axis of symmetry is x = −b/(2a) = −(100)/(2×5) = -10.
62
What is the y-intercept of y = -2x^2 − 8x + 47?
Medium
ExplanationThe y-intercept occurs at x = 0: y = -2(0)^2 + (-8)(0) + 47 = 47.
63
What is the nature of the roots of 5x^2 + 6x − 15 = 0?
Medium
ExplanationThe discriminant is b^2 − 4ac = (6)^2 − 4(5)(-15) = 336, which is positive, so the equation has two distinct real roots.
64
Solve the inequality: x^2 − 18x + 72 < 0
Medium
Explanationx^2 − 18x + 72 = (x - 6)(x - 12), with roots 6 and 12. Since the parabola opens upward, the expression is negative between the roots, giving 6 < x < 12.
Explanationx^2 − 24x + 95 = (x - 5)(x - 19), with roots 5 and 19. Since the parabola opens upward, the expression is positive outside the roots, giving x < 5 or x > 19.
82
Is x^2 − 14x + 52 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 = (-7)^2 = 49, which does not match c = 52, so it is not a perfect square trinomial.
83
What is the value of -6x^2 − 12x − 21 when x = -11?
Explanationx^2 + 5x − 50 = (x + 10)(x - 5), with roots -10 and 5. Since the parabola opens upward, the expression is negative between the roots, giving -10 < x < 5.
95
What is the product of the roots of 4x^2 + 24x − 20 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = -20/4 = -5.
96
Does the parabola y = -7x^2 − 25x + 28 open upward or downward?
Medium
ExplanationSince a = -7 is negative, the parabola opens downward.
97
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.
98
What is the product of the roots of 5x^2 − 31x − 15 = 0?
Medium
ExplanationBy Vieta's formulas, the product of the roots equals c/a = -15/5 = -3.
99
Does the parabola y = 6x^2 + 24x + 5 open upward or downward?
Medium
ExplanationSince a = 6 is positive, the parabola opens upward.
100
What is the y-intercept of y = -3x^2 − 8x + 10?
Medium
ExplanationThe y-intercept occurs at x = 0: y = -3(0)^2 + (-8)(0) + 10 = 10.