Question

If the difference of the roots is equal to the product of the roots of the equation $2x^2-(a+1)x+(a-1)=0$ then find the value of $a$.

Answer 1

Let $\alpha$ and $\beta$ be the roots of the equation

$2x^2-(a+1)x+(a-1)=0$

Given : $\alpha -\beta =\alpha \beta$

squaring on both sides,

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

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

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

We know,

$\alpha +\beta =\frac{a+1}{2}-----\left(2\right);\alpha \beta =\frac{a-1}{2}----\left(3\right)$

Substituting $\left(2\right)$ and $\left(3\right)$ in eq $\left(1\right)$

${\left(\frac{a+1}{2}\right)}^2-4\left(\frac{a-1}{2}\right)={\left(\frac{a-1}{2}\right)}^2$

Simplifying, we get.

$a=2$