Question

(x² + 1)² – x² = 0 has ______.

Options

• Four real roots

• Two real roots

• No real roots

• One real root

Answer 1

(x² + 1)² – x² = 0 has no real roots.

Explanation:

Given equation is (x² + 1)² – x² = 0

⇒ x⁴ + 1 + 2x² – x² = 0   ......[∵ (a + b)² = a² + b² + 2ab]

⇒ x⁴ + x² + 1 = 0

Let x² = y

∴ (x²)² + x² + 1 = 0

y² + y + 1 = 0

On comparing with ay² + by + c = 0, we get

a = 1, b = 1 and c = 1

Discriminant, D = b² – 4ac

= (1)² – 4(1)(1)

= 1 – 4

= – 3

Since, D < 0

∴ y² + y + 1 = 0

i.e., x⁴ + x² + 1 = 0

or (x² + 1)² – x² = 0  has no real roots.