Which of the following is not a quadratic equation?

Solution:

(d)

(a) Given that, $2(x-1)^{2}=4 x^{2}-2 x+1$

$\Rightarrow \quad 2\left(x^{2}+1-2 x\right)=4 x^{2}-2 x+1$

$\Rightarrow \quad 2 x^{2}+2-4 x=4 x^{2}-2 x+1$

$\Rightarrow \quad 2 x^{2}+2 x-1=0$

which represents a quadratic equation because it has the quadratic form $a x^{2}+b x+c=0, a \neq 0$

(b) Given that, $2 x-x^{2}=x^{2}+5$

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

which also represents a quadratic equation because it has the quadratic form $a x^{2}+b x+c=0, a \neq 0$.

(c) Given that, $(\sqrt{2} \cdot x+\sqrt{3})^{2}=3 x^{2}-5 x$

$\Rightarrow \quad 2 \cdot x^{2}+3+2 \sqrt{6} \cdot x=3 x^{2}-5 x$

$\Rightarrow \quad x^{2}-(5+2 \sqrt{6}) x-3=0$

which also represents a quadratic equation because it has the quadratic form $a x^{2}+b x+c=0, a \neq 0$

(d) Given that, $\left(x^{2}+2 x\right)^{2}=x^{4}+3+4 x^{2}$

$\Rightarrow \quad x^{4}+4 x^{2}+4 x^{3}=x^{4}+3+4 x^{2}$

$\Rightarrow \quad 4 x^{3}-3=0$

which is not of the form $a x^{2}+b x+c, a \neq 0$. Thus, the equation is not quadratic. This is a cubic equation.