# If the difference of the roots of x

Question:

If the difference of the roots of $x^{2}-p x+q=0$ is unity, then

(a) $p^{2}+4 q=1$

(b) $p^{2}-4 q=1$

(c) $p^{2}+4 q^{2}=(1+2 q)^{2}$

(d) $4 p^{2}+q^{2}=(1+2 p)^{2}$

Solution:

(b) $p^{2}-4 q=1$

Given equation: $x^{2}-p x+q=0$

Also $\alpha$ and $\beta$ are the roots of the equation such that $\alpha-\beta=1$.

Sum of the roots $=\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}=-\left(\frac{-p}{1}\right)=p$

Product of the roots $=\alpha \beta=\frac{C \text { onstant term }}{C \text { oefficient of } x^{2}}=q$

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

$\Rightarrow p^{2}-1=4 q$

$\Rightarrow p^{2}-4 q=1$