Question

If the roots of $x^2-5x+4=0$ are $p,q$, then which of the following CANNOT be the equation with the roots $\sqrt{p},\sqrt{q}$ ?

• A. $4x^4+5x^2+1=0$
• B. $x^4-5x^2-4=0$
• C. $x^4+5x^2+4=0$
• D. $4x^4-5x^2+1=0$

Answer 1

Answer: (A), (B), (C)

$x^4+5x^2+4=0$

Given that the roots of $x^2-5x+4=0$ are $p,q$
Here $a=1,b=-5,c=4$

We know that for
$a{x}^2+bx+c=0,a\ne 0$ with roots $p,q$
the transformed quadratic equation with roots $\frac{1}{\sqrt{\alpha }}\&\frac{1}{\sqrt{\beta }}$ is given by:
$c(x^2{)}^2+b(x{)}^2+a=0$

Therefore we can directly write the required equation as:
$4(x^2{)}^2-5(x{)}^2+1=0$
$\Rightarrow 4x^4-5x^2+1=0$