Answer to Question #103626 in Algebra for ea

Question

A biquadratic equation must have at least one real root. Is it true or false? give reasons.

Answer 1

A biquadratic equation is a polynomial equation with degree 4 without having degree 1 and 3 terms. Since it is saying it "must have" at least 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 $x^2=-4$

$x=±\sqrt{\smash[b]{-3}}$ x=$±\sqrt{\smash[b]{3}}i$ or $x=±\sqrt{\smash[b]{-4}}$ or $x=±2i$

Which means a biquadratic equation can have all imaginary roots.