WakeUp wrote: The discriminant of the equation $x^2-ax+b=0$ is the square of a rational number and $a$ and $b$ are integers. Prove that the roots of the equation are integers. if equation $x^2-ax+b=0$ has rational solution, for example p/q ==> q|1 and p|b ==> solutions are integers!

It can be proved that for integers $m, n$, the number $x=\sqrt[m]{n}$ is an integer or is irrational or is imaginary or if possible, doesn't exist. So, $\Delta=a^2-4b$ is the square of a rational number $x\Longrightarrow x$ is an integer and $a^2-x^2=4b$ is even$\Longrightarrow a, x$ have the same parity$\Longrightarrow a\pm x$ is even $\Longrightarrow\frac{a\pm x}{2}$ is an integer but the roots of the quadratic equation are $\frac{a\pm x}{2}$ and hence, the roots are integers as required.

$\Delta=a^2-4b$ = $\frac{p^2}{q^2}$, p,q are intengers.See that the question comes down to proving that $\frac{p}{q}$ is intenger. $a^2 -4b$ is intenger, so $\frac{p^2}{q^2}$ is intenger $\Longrightarrow q^2|p^2$. We will proof that $q|p$. Let GCD($p$,$q$)=$d$ $\Longrightarrow $ $p=dx$ and $q=dy$, GCD($x$,$y$)=1; $\frac{p^2}{q^2}$ = $\frac{x^2}{y^2}$ $\Longrightarrow$ $y=1$. So, $p=dx$ and $q=d$, so $p$ is multiple of $q$.

WakeUp wrote: The discriminant of the equation $x^2-ax+b=0$ is the square of a rational number and $a$ and $b$ are integers. Prove that the roots of the equation are integers. Isn't this directly true by RRT?

Goutham wrote: It can be proved that for integers $m, n$, the number $x=\sqrt[m]{n}$ is an integer or is irrational or is imaginary or if possible, doesn't exist. So, $\Delta=a^2-4b$ is the square of a rational number $x\Longrightarrow x$ is an integer and $a^2-x^2=4b$ is even$\Longrightarrow a, x$ have the same parity$\Longrightarrow a\pm x$ is even $\Longrightarrow\frac{a\pm x}{2}$ is an integer but the roots of the quadratic equation are $\frac{a\pm x}{2}$ and hence, the roots are integers as required. I think that because x is a rational number, it doesn't have a clear even or odd status. So, you can't assume it has the same parity as the integer a.