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$x 2 + 1 2 − x 2 = 0$ has:

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An Intiative by Sri Chaitanya

four real roots

two real roots

no real roots

one real root

answer is C.

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Detailed Solution

Given equation is,

x 2 + 1 2 − x 2 = 0

We know,

a 2 − b 2 = (a + b) (a − b)

Now,

(x 2 + 1) 2 − x 2 = 0 ⇒ (x 2 + 1 + x) (x 2 + 1 − x) = 0 ⇒ x 2 + x + 1 = 0 or x 2 − x + 1 = 0

The discriminant formula for the equation

a x^{2} + by + c = 0

is given as,

D = b^{2} - 4 ac

Now discriminant of

x 2 − x + 1 = 0

is,

⇒ b 2 − 4 a c ⇒ (− 1) 2 − 4 (1) (1) ⇒ − 3 < 0

And discriminant of

x 2 + x + 1 = 0

is,

⇒ (1) 2 − 4 (1) (1) ⇒ − 3 < 0

No real roots exists.
Therefore, no roots of given equation are real.
Hence, option 3 is correct.

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