(x² + 1)² - x² = 0 has
a. four real roots
b. two real roots
c. no real roots
d. one real root

Solution:

Given, the equation is (x² + 1)² - x² = 0

We have to find the nature of the roots of the equation.

By using algebraic identity,

(a + b)² = a² + 2ab + b²

(x² + 1)² = x⁴ + 2x² + 1

So, x⁴ + 2x² + 1 - x² = 0

By grouping,

x⁴ + 2x² - x² + 1 = 0

x⁴ + x² + 1 = 0

Let y = x²

So, y² + y + 1 = 0

Here, a = 1, b = 1 and c = 1

Discriminant = b² - 4ac = (1)² - 4(1)(1)

= 1 - 4

= -3 < 0

A quadratic equation ax² + bx + c = 0 has no real roots when the discriminant of the equation is less than zero.

Therefore, the equation has no real roots.

✦ Try This: (x² + 2)² - x² + 2x = 0 has

1. four real roots
2. two real roots
3. no real roots
4. one real root

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 4

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NCERT Exemplar Class 10 Maths Exercise 4.1 Problem 11

(x² + 1)² - x² = 0 has, a. four real roots, b. two real roots, c. no real roots, d. one real root

Summary:

The equation (x² + 1)² - x² = 0 has no real roots

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