Let $\alpha$ and $\beta$ be the roots of a quadratic equation $4x^2-(5p+1)x+5=0$.If $\beta=1+\alpha,$then find the integral value of $p.$
Sum of roots$=\alpha+\beta=\frac{5p+1}{4}$
Given $\beta=1+\alpha,$so $\alpha-\beta=-1$
Adding the two equations,
$$\alpha=\frac{5p-3}{8}$$
As $\alpha$ is a root of the equation $4x^2-(5p+1)x+5=0$.So $4\alpha^2-(5p+1)\alpha+5=0$
$$4(\frac{5p-3}{8})^2-(5p+1)(\frac{5p-3}{8})+5=0$$
$$5p^2+2p-19=0$$
But when i solve this equation,i do not get integer values of $p.$
But the answer given is that integer value of $p$ is $3.$ I do not know where i am wrong?