Finding a quadratic equation with roots $x=α^2$ and $x=β^2$

Question

I'm struggling on a question and have no idea where to start.

The quadratic equation $y=ax^2+bx+c$ has roots $x=\alpha$ and $x=\beta$. Find an equation that has roots $x=\alpha^2$ and $x=\beta^2$.

I was just wondering if anyone could provide any knowledge so I can start this question?

Answer 1

Assuming you want to express the quadratic with roots $\alpha^2$ and $\beta^2$ in terms of $a,b,c$, from Vieta's:

$$\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta=\frac{b^2}{a^2}-\frac{2c}{a}$$

$$\alpha^2\beta^2=\frac{c^2}{a^2}$$

Can you end it from here?

Answer 2

Hints:

• Can you express the sum $\alpha+\beta$ and the product $\alpha\beta$ of the original roots in terms of the original coefficients $a,b,c$?
• Can you express the sum $\alpha^2+\beta^2$ and the product $\alpha^2\beta^2$ of the new roots in terms of the sum $\alpha+\beta$ and the product $\alpha\beta$ of the original roots?
• Can you express the new coefficients $a',b',c'$ in terms of the sum $\alpha^2+\beta^2$ and the product $\alpha^2\beta^2$ of the new roots?

Answer 3

Apply the Graeffe root-squaring method.

$ax^2+bx+c=0$

$ax^2+c=-bx$ separating even and odd degree terms

$a^2x^4+2acx^2+c^2=b^2x^2$ after squaring

$\color{blue}{a^2(x^2)^2+(2ac-b^2)(x^2)+c^2=0}$