Question

Let $\alpha ,\beta$ be the roots of the equation $a{x}^2+bx+c=0$. Then find the quadratic equation whose roots are $\alpha +\beta$ and $\alpha .\beta$.

Answer 1

Given $\alpha ,\beta$ are roots of the quadrite equation $a{x}^2+bx+e=0$

Then from the relation between roots & co-efficients we get,

$\alpha +\beta =-\frac{b}{a}--------\left(1\right)$ & $\alpha \beta =\left(c/a\right)-------\left(2\right).$

Now, we are to find a quadratic equation whose roots are $(\alpha +\beta )$ and $\left(\alpha \beta \right)$ is.

$x-(\alpha +\beta )x-\left(\alpha \beta \right)=0$

on, $(x+\frac{b}{a})(x-\frac{c}{a})=0$

on, $x^2+2\frac{(b-c)}{a}-\frac{bc}{a^2}=0$

on, $a^2{x}^2+ax(b-c)-bc=0$.

This is the required quadratic equation.