Question

Which of the following is not a quadratic equation?

• A. $(\sqrt{2}x+\sqrt{3}{)}^2=3x^2-5x$
• B. $2x-x^2=x^2+5$
• C. $(x^2+2x{)}^2=x^4+3+4x^2$
• D. $x^2+5x=10$

Answer 1

The correct option is C

$(x^2+2x{)}^2=x^4+3+4x^2$

Consider $(\sqrt{2}x+\sqrt{3}{)}^2=3x^2-5x.$
$\Rightarrow 2x^2+3+2\sqrt{6}x=3x^2-5x\Rightarrow x^2-(5+2\sqrt{6})x-3=0,$
which is a quadratic equation as it is in the quadratic form $a{x}^2+bx+c=0,a\ne 0.$

Consider $2x-x^2=x^2+5.$
$\Rightarrow 2x^2-2x+5=0,$
a quadratic equation as it is in the quadratic form $a{x}^2+bx+c=0,a\ne 0.$

Let us take $(x^2+2x{)}^2=x^4+3+4x^2.$
$\Rightarrow x^4+4x^2+4x^3=x^4+3+4x^2\Rightarrow 4x^3-3=0,$
a cubic equation as it is not of the form $a{x}^2+bx+c=0,a\ne 0.$ Thus, the equation is not quadratic.

Lastly, let us consider $x^2+5x=10.$ This represents a quadratic equation as it is in the quadratic form $a{x}^2+bx+c=0,a\ne 0.$