Question

If $\alpha ,\beta$ are the roots of the equation $4x^2+6x+3=0$ find the equation whose roots are ${\alpha }^2$ & ${\beta }^2$.

Answer 1

Given $\alpha$ and $\beta$ are the roots of the equation $4x^2+6x+3=0$

Sum of the roots$=\alpha +\beta =\frac{-6}{4}=\frac{-3}{2}$

Product of the roots$=\alpha \beta =\frac{3}{4}$

${\alpha }^2+{\beta }^2={(\alpha +\beta )}^2-2\alpha \beta$

${\alpha }^2+{\beta }^2={\left(\frac{-3}{2}\right)}^2-2\frac{3}{4}$

${\alpha }^2+{\beta }^2=\frac{9}{4}-\frac{6}{4}=\frac{9-6}{4}=\frac{3}{4}$

$\therefore {\alpha }^2+{\beta }^2=\frac{3}{4}$

and ${\alpha }^2{\beta }^2={\left(\alpha \beta \right)}^2={\left(\frac{3}{4}\right)}^2=\frac{9}{16}$

The required equation is $x^2-({\alpha }^2+{\beta }^2)x+{\alpha }^2{\beta }^2=0$

$x^2-\frac{3}{4}x+\frac{9}{16}=0$

or $16x^2-12x+9=0$