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Find the value of x.
$2 x x − 5 2 + 10 x x − 5 − 24 = 0 x ≠ 5$

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7 or15

4 or 25

4 or 15

3 or 5

answer is A.

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Detailed Solution

The given equation is

2 x x − 5 2 + 10 x x − 5 − 24 = 0 x ≠ 5.

The quadratic equation is

a x 2 + b x + c = 0

and the roots of the equation is given by quadratic formula that is

x = − b ± D 2 a

where

D = b 2 − 4 a c

is called discriminant.
Simplifying the given equation to write in quadratic form,

⇒ 2 x x − 5 2 + 10 x x − 5 − 24 = 0 ⇒ 4 x 2 + 10 x x − 5 − 24 x − 5 2 x − 5 2 = 0 ⇒ 4 x 2 + 10 x 2 − 50 x − 24 x 2 + 25 − 10 x = 0 ⇒ 14 x 2 − 50 x − 24 x 2 − 600 + 240 x = 0

Comparing the equation (1) with the standard quadratic equation

a x 2 + b x + c = 0

to find the value of a, b and c.
Here,

a = 1

b = − 19

and

c = 60

Using the formula

D = b 2 − 4 a c

to find the discriminant D,

⇒ D = b 2 − 4 a c ⇒ D = − 19 2 − 4160 ⇒ D = 361 − 240 ⇒ D = 121

Now, using the quadratic formula to find the value of x.

⇒ x = − b ± D 2 a ⇒ x = − − 19 ± 121 21 ⇒ x = 19 ± 11 2 ⇒ x = 82, 302 ⇒ x = 4, 15

Therefore, the values of x is 4 or 15.
Hence, option (1) is correct.

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