Question

Which of the following is a quadratic equation?

Options

• x² + 1 = (2 – x)² + 3

• 2x² + 3 = (5 + x) (2x – 3)

• x³ – x² = (x – 1)³

• none of these

Answer 1

x³ – x² = (x – 1)³

Explanation:

A quadratic equation is one that can be written in the form ax² + bx + c = 0 with a ≠ 0.

(a) x² + 1 = (2 – x)² + 3

⇒ x² + 1 = x² – 4x + 7

⇒ 1 = –4x + 7   ...(Linear)

(b) 2x² + 3 = (5 + x)(2x – 3)

⇒ 2x² + 3 = 2x² + 7x – 15 

⇒ 3 = 7x – 15   ...(Linear)

(c) x³ – x² = (x – 1)³

⇒ x³ – x² = x³ – 3x² + 3x – 1

⇒ 0 = –2x² + 3x – 1

⇒ 2x² – 3x + 1 = 0   ...(Quadratic)

Since x³ – x² = (x – 1)³ simplifies to 2x² – 3x + 1 = 0, it is a quadratic equation.