Answer to Question #104154 in Algebra for Maria

Question

A biquadratic equation must have at least one real root is it true or false ? Give reason?

Answer 1

A biquadratic equation must have at least one real root is it true or false ?

A biquadratic equation is a polynomial equation with degree 4 without having degree 1 and 3 terms. Since it is saying it "must have" atleast one real root, it makes the statement false.

To prove this I will provide a counter example.

Consider a biquadratic equation

$x^4+7x^2+12=0$

For finding roots , let $u=x^2$

Given equation will rewritten as

$u^2+7u+12=0$

$u^2+3u+4u+12=0$

$u(u+3)+4(u+3)=0$

$(u+3)(u+4)=0$

i.e. $u= -3$ or $u = -4$

since $u=x^2$

$x^2=-3$ or $x^2=-4$

i.e. $x=\pm \sqrt{-3}$ or $x=\pm \sqrt{-4}$

i.e $x=\pm \sqrt{3}i$ or $x=\pm 2i$

Which means a biquadratic equation can have all imaginary roots.