Question

If $\alpha$ and $\beta$ are zeros of a quadratic polynomial such that $\alpha +\beta =24$ and $\alpha -\beta =8$ find a quadtratic polynomial having $\alpha$ and $\beta$ as it's zeros.

Answer 1

$\alpha +\beta =24$

$\alpha -\beta =8$

${\left(\alpha -\beta \right)}^2={\left(\alpha +\beta \right)}^2-4\alpha \beta =8^2$

$or,{24}^2-4\alpha \beta =64$

$or,4\alpha \beta =576-64=512$

$or,\alpha \beta =128$

$\Rightarrow x^2-(\alpha +\beta )x+\alpha \beta =0$

$or,x^2-24x+128=0$

Hence $x^2-24x+128=0$ is the quadratic polynomial for the required condition