Question

If $\alpha$ and $\beta$ are zeroes of the polynomial $p\left(x\right)=6x^2-5x+k$ such that $\alpha -\beta =\frac{1}{6}$, find the value of $k$.

Answer 1

Given $\alpha$ and $\beta$ are zeros of the polynomial $p\left(x\right)=6x^2-5x+k$,
From the given equation
Sum of the roots : $\alpha +\beta =\frac{-b}{a}=\frac{5}{6}$
Product of the roots: $\alpha \beta =\frac{c}{a}=\frac{k}{6}$

To,find the value of $k$,

$\Rightarrow \alpha -\beta =\frac{1}{6}$

$\Rightarrow {\alpha }^2+{\beta }^2-2\alpha \beta =\frac{1}{36}$

$\Rightarrow {(\alpha +\beta )}^2-4\alpha \beta =\frac{1}{36}$

$\Rightarrow \frac{25}{36}-\frac{4k}{36}=\frac{1}{36}$

$\Rightarrow 25-4k=1$

$\Rightarrow k=6$