Question

Which of the following is a quadratic equation?

Options

• `x^2 + 2x + 1 = (4 - x)^2 + 3`

• `-2x^2 = (5 - x)(x - 2/5)`

• `(k + 1)x^2 + 3/2x` = 7, where k = –1

• `x^3 - x^2 = (x - 1)^3`

Answer 1

x³ – x² = (x – 1)³

Explanation:

The standard form of a quadratic equation is given by,

ax² + bx + c = 0, a ≠ 0

(A) Given, x² + 2x + 1 = (4 – x)² + 3

x² + 2x + 1 = 16 – 8x + x² + 3

10x – 18 = 0

which is not a quadratic equation.

(B) Given, –2x² = `(5 - x)(2x - 2/5)`

–2x² = `10x - 2x^2 - 2 + (2x)/5`

50x + 2x – 10 = 0

52x – 10 = 0

which is not a quadratic equation.

(C) Given, `(k + 1)x^2 + 3/2x` = 7, where k = –1

`(-1 + 1)x^2 + 3/2x` = 7

3x – 14 = 0

which is not a quadratic equation.

(D) Given, x³ – x² = (x – 1)³

x³ – x² = x³ – 3x²(1) + 3x(1)² – (1)³  ...[ ∵ (a – b)³ = a³ – b³ + 3ab² – 3a²b]

x³ – x² = x³ – 3x² + 3x – 1

–x² + 3x² – 3x + 1 = 0

2x² – 3x + 1 = 0

which represents a quadratic equation.