WAEC 2010 · Paper 2 · Q4

(a)
Find the equation whose roots are the squares of the roots of the equation $x^2 - mx + n = 0$ .
Show the answer
$x^2 - (m^2 - 2n)x + n^2 = 0$

Worked solution (try it first)

Let the roots of $x^2 - mx + n = 0$ be $\alpha$ and $\beta$ : $\alpha + \beta = m$ and $\alpha\beta = n$ .
The new roots are $\alpha^2$ and $\beta^2$ .
Their sum: $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$
$= m^2 - 2n$ .
Their product: $\alpha^2\beta^2 = (\alpha\beta)^2 = n^2$ .
Use $x^2 - (\text{sum})x + \text{product} = 0$ : $x^2 - (m^2 - 2n)x + n^2 = 0$ .