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Find the roots of the quadratic equation: 2×2 + 3x – 9 = 0 ?
A. -3/2, -3
B. 3/2, 3
C. 3/2, -3
D. 3, -3/2Explanation:
2×2 + 6x – 3x – 9 = 0
2x(x + 3) – 3(x + 3) = 0
(x + 3)(2x – 3) = 0
=> x = -3 or x = 3/2.Roots of the equation 3×2 – 12x + 10 = 0 are ?
A. real and equal
B. irrational and unequal
C. rational and unequal
D. complexIf the roots of a quadratic equation are 20 and -7, then find the equation ?
A. x2 + 13x + 140 = 0
B. x2 – 13x – 140 = 0
C. x2 + 13x – 140 = 0
D. x2 – 13x + 140 = 0Explanation:
Any quadratic equation is of the form
x2 – (sum of the roots)x + (product of the roots) = 0 —- (1)
where x is a real variable. As sum of the roots is 13 and product of the roots is -140, the quadratic equation with roots as 20 and -7 is: x2 – 13x – 140 = 0.Sum and the product of the roots of the quadratic equation x(power 2) + 20x + 3 = 0 are ?
A. -20, 3
B. -10, -3
C. 10, 3
D. -10, 3If the roots of the equation 2×2 – 5x + b = 0 are in the ratio of 2:3, then find the value of b ?
A. 12
B. 43
C. 3
D. 5Let the roots of the equation 2a and 3a respectively.
2a + 3a = 5a = -(- 5/2) = 5/2 => a = 1/2
Product of the roots: 6a2 = b/2 => b = 12a2
a = 1/2, b = 3.Sum of the squares of two consecutive positive integers exceeds their product by 91. Find the integers ?
A. 11, 12
B. 12, 13
C. 9, 10
D. 10, 11Explanation:
Let the two consecutive positive integers be x and x + 1
x2 + (x + 1)2 – x(x + 1) = 91
x2 + x – 90 = 0
(x + 10)(x – 9) = 0 => x = -10 or 9.
As x is positive x = 9
Hence the two consecutive positive integers are 9 and 10.One root of the quadratic equation x2 – 12x + a = 0, is thrice the other. Find the value of a ?
A. 12
B. 45
C. 27
D. 29Explanation:
Let the roots of the quadratic equation be x and 3x.
Sum of roots = -(-12) = 12
a + 3a = 4a = 12 => a = 3
Product of the roots = 3a2 = 3(3)2 = 27.Sm of the square of the three consecutive even natural numbers is 1460. Find the numbers ?
A. 22, 24, 26
B. 24, 26, 28
C. 20, 22, 24
D. 18, 20, 22Explanation:
Three consecutive even natural numbers be 2x – 2, 2x and 2x + 2.
(2x – 2)2 + (2x)2 + (2x + 2)2 = 1460
4×2 – 8x + 4 + 4×2 + 8x + 4 = 1460
12×2 = 1452 => x2 = 121 => x = ± 11
As the numbers are positive, 2x > 0. Hence x > 0. Hence x = 11.
Required numbers are 20, 22, 24.If a and b are the roots of the equation x2 – 9x + 20 = 0, find the value of a2 + b2 + ab ?
A. 61
B. 65
C. -21
D. 1Explanation:
a2 + b2 + ab = a2 + b2 + 2ab – ab
i.e., (a + b)2 – ab
from x2 – 9x + 20 = 0, we have
a + b = 9 and ab = 20. Hence the value of required expression (9)2 – 20 = 61.Find the value of a/b + b/a, if a and b are the roots of the quadratic equation x2 + 8x + 4 = 0 ?
A. 14
B. 3
C. 24
D. 21Explanation:
a/b + b/a = (a2 + b2)/ab = (a2 + b2 + a + b)/ab
= [(a + b)2 – 2ab]/ab
a + b = -8/1 = -8
ab = 4/1 = 4
Hence a/b + b/a = [(-8)2 – 2(4)]/4 = 56/4 = 14.