Question

Which of the following is not a quadratic equation?

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

Hint:

Always fully expand and simplify the expressions on both sides of the equation. An equation might look quadratic at first, but the \(x^2\) terms might cancel out.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A quadratic equation has the standard form \(ax^2 + bx + c = 0\) with \(a \neq 0\). We need to simplify each equation and find the one where the highest power of \(x\) is not 2.

Step 2: Detailed Explanation:
(A) \(5x - x^2 = x^2 + 3 $\Rightarrow$ 2x^2 - 5x + 3 = 0\). This is a quadratic equation.
(B) \(x^3 - x^2 = (x-1)^3 $\Rightarrow$ x^3 - x^2 = x^3 - 3x^2(1) + 3x(1)^2 - 1^3 $\Rightarrow$ x^3 - x^2 = x^3 - 3x^2 + 3x - 1\). The \(x^3\) terms cancel. \(\Rightarrow 2x^2 - 3x + 1 = 0\). This is a quadratic equation.
(C) \((x+3)^2 = 3(x^2 - 5) $\Rightarrow$ x^2 + 6x + 9 = 3x^2 - 15 $\Rightarrow$ 2x^2 - 6x - 24 = 0\). This is a quadratic equation.
(D) \((\sqrt{2}x + 3)^2 = 2x^2 + 5 $\Rightarrow$ (\sqrt{2}x)^2 + 2(\sqrt{2}x)(3) + 3^2 = 2x^2 + 5 $\Rightarrow$ 2x^2 + 6\sqrt{2}x + 9 = 2x^2 + 5\). The \(2x^2\) terms cancel out, leaving \(6\sqrt{2}x + 4 = 0\). This is a linear equation, not a quadratic equation.

Step 3: Final Answer:
The equation that is not a quadratic equation is \((\sqrt{2}x + 3)^2 = 2x^2 + 5\).