Ram and Shyam solve a quadratic equation. Ram makes a mistake in the constant term and finds the roots as $5,-3$. Shyam makes a mistake in the co-efficient of $x$ and gets the roots as $1,-3$. The correct roots are:

Let the standard quadratic equation be written as:
$$x^2-(\text{Sum of roots})x+(\text{Product of roots})=0$$

1. Ram's calculation:

* Ram makes a mistake in the constant term, which means his coefficient of $x$ (and thus his Sum of roots) is correct.

$$\text{Correct Sum of roots}=5+(-3)=2$$

2. Shyam's calculation:

* Shyam makes a mistake in the coefficient of $x$, which means his constant term (and thus his Product of roots) is correct.

$$\text{Correct Product of roots}=1\times (-3)=-3$$

3. Forming the correct equation:
Substituting the correct sum and product back into the standard quadratic form:

$$x^2-2x+(-3)=0$$

$$x^2-2x-3=0$$

Factoring the quadratic equation:

$$x^2-3x+x-3=0$$

$$x(x-3)+1(x-3)=0$$

$$(x-3)(x+1)=0$$

Thus, the correct roots are $3$ and $-1$.