Question

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f\left(x\right)=x^2-5x+k$ such that $\alpha -\beta =1$, then the value of $k$ is ___________.

• A. $-6$
• B. $5$
• C. $-5$
• D. $6$

Answer 1

Answer: (D)

$6$

Since $\alpha ,\beta$ are the zeroes of the polynomial, they are also the roots of the equation $x^2-5x+k=0$

Sum of the roots $=\alpha +\beta =-b/a=5$ $\to \left(I\right)$

We know from the question that $\alpha -\beta =1$ $\to \left(II\right)$

$\left(I\right)+\left(II\right)\Rightarrow 2\alpha =6$

$\Rightarrow \alpha =3$

$\Rightarrow \beta =2$ (by substituting for $\alpha$ in $\left(I\right)$)

$\therefore \alpha \beta =2\times 3=6$ $\to \left(III\right)$

Produuct of the roots $=\alpha \beta =c/a=k\to \left(IV\right)$

$\therefore k=6$ (from $\left(III\right),\left(IV\right)$)

Hence, the answer is option $D$.