Linear Quadratic Equations Competitive Exams
50 questions · Mathematics
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Question 1 of 50
The roots of 2x^2 - 8 = 0 are
Explanationx^2 = 4, so x = plus or minus 2.
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Question 2 of 50
The roots of (x - 3)(x + 4) = 0 are
ExplanationSet each factor to zero.
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Question 3 of 50
Solve: 2x - 3 = -9
Explanation2x = -6, so x = -3.
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Question 4 of 50
The sum of two numbers is 25 and their difference is 7. The larger number is
ExplanationLarger = (25 + 7)/2 = 16; smaller = 9.
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Question 5 of 50
The product of two consecutive positive integers is 72. The smaller integer is
Explanationx(x + 1) = 72 gives x^2 + x - 72 = 0, so x = 8.
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Question 6 of 50
If x + y = 10 and x - y = 4, then y =
ExplanationAdding gives 2x = 14, so x = 7 and y = 3.
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Question 7 of 50
Solve: x/3 - 1 = 2
Explanationx/3 = 3, so x = 9.
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Question 8 of 50
If 3x + 2y = 16 and x + y = 6, then y =
Explanationy = 6 - x, so 3x + 12 - 2x = 16, giving x = 4 and y = 2.
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Question 9 of 50
The sum of the roots of 2x^2 - 7x + 3 = 0 is
ExplanationSum = -b/a = 7/2.
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Question 10 of 50
The area of a rectangle is 48 cm^2 and its length is 2 cm more than its breadth. The breadth is
Explanationb(b + 2) = 48 gives b^2 + 2b - 48 = 0, so b = 6.
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Question 11 of 50
Using the quadratic formula, the roots of x^2 + 4x + 1 = 0 are
Explanationx = (-4 plus or minus root(16 - 4))/2 = -2 plus or minus root 3.
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Question 12 of 50
The sum of two consecutive integers is 41. The smaller integer is
Explanationx + (x + 1) = 41, so 2x = 40 and x = 20.
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Question 13 of 50
The square of a positive number exceeds the number by 56. The number is
Explanationx^2 - x = 56 gives x^2 - x - 56 = 0, so (x - 8)(x + 7) = 0 and x = 8.
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Question 14 of 50
The roots of x^2 - 3x - 10 = 0 are
ExplanationFactorise: (x - 5)(x + 2) = 0.
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Question 15 of 50
The roots of x^2 + 7x + 12 = 0 are
ExplanationFactorise: (x + 3)(x + 4) = 0.
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Question 16 of 50
Solve: 0.5x + 2 = 6
Explanation0.5x = 4, so x = 8.
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Question 17 of 50
The roots of x^2 - 4x = 0 are
Explanationx(x - 4) = 0.
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Question 18 of 50
Solve: 3x - 7 = 11
Explanation3x = 18, so x = 6.
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Question 19 of 50
If 2 is a root of x^2 - 5x + k = 0, then k =
ExplanationPut x = 2: 4 - 10 + k = 0, so k = 6.
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Question 20 of 50
The roots of 2x^2 - 3x + 5 = 0 are
ExplanationD = 9 - 40 = -31 < 0.
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Question 21 of 50
The product of the roots of 3x^2 + 5x - 2 = 0 is
ExplanationProduct = c/a = -2/3.
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Question 22 of 50
The roots of x^2 + 5x + 2 = 0 are
ExplanationD = 25 - 8 = 17, which is positive but not a perfect square.
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Question 23 of 50
The roots of x^2 - 2x - 15 = 0 are
ExplanationFactorise: (x - 5)(x + 3) = 0.
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Question 24 of 50
Solve: (2x + 1)/3 = 5
Explanation2x + 1 = 15, so x = 7.
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Question 25 of 50
The value of k for which x^2 + kx + 9 = 0 has equal roots is
ExplanationEqual roots need D = 0: k^2 - 36 = 0, so k = plus or minus 6.
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Question 26 of 50
A number is doubled and then increased by 7 to give 25. The number is
Explanation2x + 7 = 25, so x = 9.
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Question 27 of 50
Solve: 5x = 3x + 10
Explanation2x = 10, so x = 5.
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Question 28 of 50
Solve: -2x + 8 = 0
Explanation-2x = -8, so x = 4.
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Question 29 of 50
A father is 3 times as old as his son. The sum of their ages is 48. The son's age is
Explanationx + 3x = 48, so x = 12.
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Question 30 of 50
Solve: (x - 1)/2 + (x + 1)/3 = 4
ExplanationMultiply by 6: 3(x - 1) + 2(x + 1) = 24, so 5x - 1 = 24 and x = 5.
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Question 31 of 50
The discriminant of x^2 - 6x + 9 = 0 is
ExplanationD = b^2 - 4ac = 36 - 36 = 0, so the roots are real and equal.
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Question 32 of 50
Solve: 3(x + 2) = 2(x + 5)
Explanation3x + 6 = 2x + 10, so x = 4.
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Question 33 of 50
Solve: 4(x - 2) = 12
Explanationx - 2 = 3, so x = 5.
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Question 34 of 50
If a and b are the roots of x^2 - 5x + 6 = 0, then a^2 + b^2 =
Explanation(a + b)^2 - 2ab = 25 - 12 = 13.
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Question 35 of 50
The root of x^2 + 6x + 9 = 0 is
Explanation(x + 3)^2 = 0 gives one repeated root x = -3.
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Question 36 of 50
If 2x + 3y = 12 and x - y = 1, then x =
Explanationx = y + 1, so 2y + 2 + 3y = 12, giving y = 2 and x = 3.
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Question 37 of 50
Solve: 7x - 4 = 3x + 12
Explanation4x = 16, so x = 4.
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Question 38 of 50
The roots of x^2 + 2x - 8 = 0 are
ExplanationFactorise: (x - 2)(x + 4) = 0.
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Question 39 of 50
The roots of x^2 - x - 12 = 0 are
ExplanationFactorise: (x - 4)(x + 3) = 0.
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Question 40 of 50
The sum of two consecutive even integers is 54. The smaller one is
Explanationx + (x + 2) = 54 gives x = 26.
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Question 41 of 50
The sum of a number and its reciprocal is 10/3. The number is
Explanation3x^2 - 10x + 3 = 0, so (3x - 1)(x - 3) = 0.
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Question 42 of 50
The quadratic equation whose roots are 2 and 5 is
Explanationx^2 - (sum)x + product = x^2 - 7x + 10 = 0.
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Question 43 of 50
Solve: x/4 + 3 = 7
Explanationx/4 = 4, so x = 16.
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Question 44 of 50
The roots of x^2 - 9 = 0 are
Explanationx^2 = 9, so x = plus or minus 3.
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Question 45 of 50
The roots of 2x^2 - 5x - 3 = 0 are
ExplanationFactorise: (2x + 1)(x - 3) = 0.
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Question 46 of 50
Solve: 2x + 5 = 13
Explanation2x = 13 - 5 = 8, so x = 4.
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Question 47 of 50
If 2x - y = 1 and x + y = 5, then xy =
ExplanationAdding gives 3x = 6, so x = 2 and y = 3; xy = 6.
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Question 48 of 50
The roots of x^2 - 5x + 6 = 0 are
ExplanationFactorise: (x - 2)(x - 3) = 0.
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Question 49 of 50
The perimeter of a rectangle is 36 cm and its length is 2 cm more than its breadth. The length is
Explanation2(b + b + 2) = 36 gives b = 8, so the length = 10 cm.
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Question 50 of 50
Solve: x + x/2 = 9
Explanation3x/2 = 9, so x = 6.
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