Question

If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $x^2+5x-k$ and $\alpha -\beta =1$, then find the value of $k$.

Answer 1

Given the quadratic expression $x^2-5x+k$

Here, $a=1,b=-5$ and $c=k$

We know by rule of sum and product of roots that,

$\alpha +\beta =-\frac{b}{a}=-\frac{-5}{1}=5$

$\alpha \beta =\frac{c}{a}=\frac{k}{7}=k$

Given that $\alpha -\beta =1$

Squaring both sides, we get,

$(\alpha -\beta {)}^2=1$

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

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

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

$\Rightarrow (5{)}^2-4k=1$

$\therefore k=6$