Question

If $\alpha$ and $\beta$ are zeroes of the polynomial $f\left(x\right)=x^2-x-k$, such that $\alpha -\beta =9$
find k.

Answer 1

Given that $\alpha$ and $\beta$ are the roots of the polynomial $f\left(x\right)=x^2-x-k$ such that $\alpha -\beta =9.$

To find: Value of $k.$

By using the relation between zeroes and the coefficient of a quadratic polynomial,

$\alpha +\beta =1$

$\alpha \cdot \beta =-k$

Squaring on both sides of the equation $\alpha -\beta =9$, gives,

${\alpha }^2+{\beta }^2-2\alpha \beta =81\Rightarrow {(\alpha +\beta )}^2-4\alpha \beta =81\Rightarrow 1+4k=81\Rightarrow 4k=80\Rightarrow k=20$

Hence, value of $k$ is $20.$