Question

Which of the following is a quadratic equation?

• A. \( x^2 - 3\sqrt{x} + 2 = 0 \)
• B. \( x + \frac{1}{x} = x^2 \)
• C. \( x^2 + \frac{1}{x^2} = 5 \)
• D. \( 2x^2 - 5x = (x-1)^2 \)

Hint:

A quadratic equation must have the highest exponent of \( x \) equal to 2 and should not contain fractional or radical terms.

Answer 1

The Correct Option is D

Step 1: Definition of a quadratic equation A quadratic equation is in the form: \[ ax^2 + bx + c = 0 \] where \( a, b, c \) are constants, and \( a \neq 0 \). Step 2: Analyze each option - (A) \( x^2 - 3\sqrt{x} + 2 = 0 \) contains \( \sqrt{x} \), making it non-quadratic. - (B) \( x + \frac{1}{x} = x^2 \) contains \( \frac{1}{x} \), making it non-quadratic. - (C) \( x^2 + \frac{1}{x^2} = 5 \) contains \( \frac{1}{x^2} \), making it non-quadratic. - (D) \( 2x^2 - 5x = (x-1)^2 \) expands to \( 2x^2 - 5x = x^2 - 2x + 1 \), which simplifies to: \[ x^2 - 3x - 1 = 0 \] which is a quadratic equation.