Question

Which of the following is not a quadratic equation?

• A. $2(x-1)=4x^2-2x+1$
• B. $2x-x^2=x^2+5$
• C. ${(\sqrt{2}x+\sqrt{3})}^2+x^2=3x^2-5x$
• D. ${(x^2+2x)}^2=x^4+3+4x^3$

Answer 1

The correct option is C

${(\sqrt{2}x+\sqrt{3})}^2+x^2=3x^2-5x$

Explanation for the correct option:

The quadratic equations are the polynomial equations of degree $2$ in one variable of type $f\left(x\right)=a{x}^2+bx+c$ where $a,b,c,\in R$ and $a\ne 0$. It is the general form of a quadratic equation where $\text{'}a\text{'}$ is called the leading coefficient and $\text{'}c\text{'}$ is called the absolute term of$f\left(x\right)$.

Option (C)

$$\begin{gathered} (\sqrt{2}x+\sqrt{3}){}^2+x^2=3x^2-5x \\ \Rightarrow {{\left(\sqrt{2}x\right)}^2+\left(\sqrt{3}\right)}^2+2\times \sqrt{2}x\times \sqrt{3}+x^2=3x^2-5x \\ \Rightarrow {2x}^2+3+2\sqrt{6}x+x^2=3x^2-5x \\ \therefore (5+2\sqrt{6})x-3=0 \end{gathered}$$

Here the coefficient of $x^2$ is zero. So, it is not a quadratic equation.

Explanation for the incorrect options.

Option (A)

$$\begin{gathered} 2(x-1)=4x^2-2x+1 \\ \Rightarrow 2x-2=4x^2-2x+1 \\ \therefore 4x^2-4x+3=0 \end{gathered}$$

Here the coefficient of $x^2$ is non-zero. So, it's a quadratic equation.

Option (B)

$$\begin{gathered} 2x-x^2=x^2+5 \\ \Rightarrow 2x^2-2x+5=0 \end{gathered}$$

Here the coefficient of $x^2$ is non-zero. So, it's a quadratic equation.

Option (D)

$$\begin{gathered} {(x^2+2x)}^2=x^4+3+4x^3 \\ \Rightarrow {{\left(x^2\right)}^2+\left(2x\right)}^2+2.x^2.2x=x^4+3+4x^3 \\ \Rightarrow {x^4+4x}^2+4x^3=x^4+3+4x^3 \\ \therefore {4x}^2+3=0 \end{gathered}$$

Here the coefficient of $x^2$ is non-zero. So, it's a quadratic equation.

Hence, the correct choice is “Option C”.