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Which of the following equations has two distinct real roots?

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2 x 2 − 32 x + 94 = 0

x 2 + x − 5 = 0

x 2 + 3 x + 22 = 0

5 x 2 − 3 x + 1 = 0

answer is B.

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Detailed Solution

An equation of the form

a x 2 + b x + c = 0, a ≠ 0

is called quadratic equation in one variable x, where a, b, c are constants.
We know that, the condition for real roots of the quadratic equation is

D = b 2 − 4 a c ≥ 0.

Let us find the discriminant for all the options given.
Consider option (1)

2 x 2 − 32 x + 94 = 0

Here,

a = 2, b = − 32, c = 94 .

D = b 2 − 4 a c = (− 32) 2 − 4 (2) 94 = 18 − 18 = 0

Thus, the equation

2 x 2 − 32 x + 94 = 0

has equal and real roots.
Now consider option (2)

x 2 + x − 5 = 0

Here,

a = 1, b = 1, c = − 5.

D = b 2 − 4 a c = 12 − 4 (1) (− 5) = 1 + 20 = 21 D > 0

Thus, the equation

x 2 + x − 5 = 0

has two distinct real roots.
Now option (3)

x 2 + 3 x + 22 = 0

Here,

a = 1, b = 3, c = 22 .

D = b 2 − 4 a c = 32 − 4 (1) (22) = 9 − 82 D < 0

Thus, the equation

x 2 + 3 x + 22 = 0

has no real roots because

D < 0

. Now option (4)

5 x 2 − 3 x + 1 = 0

Here,

a = 5, b = − 3, c = 1.

D = b 2 − 4 a c = (− 3) 2 − 4 (5) (1) = 9 − 20 = − 11

Thus, the equation

5 x 2 − 3 x + 1 = 0

has no real roots because

D < 0

.
The equation

x 2 + x − 5 = 0

has two distinct real roots.
The correct option is (2).

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