Question

For the equation $2x^2+(a+1)x+(a-1)=0$, the difference between the roots is equal to their product. Then the value of $a$ is

• A. $1$
• B. $2$
• C. $4$
• D. $3$

Answer 1

The correct option is B $2$

Let the roots be $\alpha ,\beta .$
Using the relation between roots and coefficients, we have
$$\begin{array}{ccc}& \alpha +\beta =\frac{a+1}{2};\alpha \beta =\frac{a-1}{2}& \\ & \text{Using}(\alpha -\beta {)}^2=(\alpha +\beta {)}^2-2\alpha \beta & \Rightarrow {\left(\frac{a-1}{2}\right)}^2={\left(\frac{a+1}{2}\right)}^2-2(a-1)\\ & \Rightarrow \frac{(a-1{)}^2}{4}=\frac{(a+1{)}^2}{4}-\frac{8(a-1)}{4}& \\ & \Rightarrow (a-1{)}^2=(a+1{)}^2-8(a-1)& \\ & \Rightarrow 4a=8\Rightarrow a=2& \end{array}$$