Write two numbers that multiply to the value on top and add to the value on bottom.

Step 1 ：Let's denote the two numbers as \(x\) and \(y\). According to the problem, we have two equations: \(xy = a\) and \(x + y = b\), where \(a\) is the value on top and \(b\) is the value on bottom.

Step 2 ：We can solve this system of equations by expressing \(y\) from the first equation: \(y = \frac{a}{x}\) and substituting it into the second equation: \(x + \frac{a}{x} = b\).

Step 3 ：This equation can be transformed into a quadratic equation: \(x^2 - bx + a = 0\).

Step 4 ：We can solve this quadratic equation using the quadratic formula: \(x = \frac{b \pm \sqrt{b^2 - 4a}}{2}\).

Step 5 ：The two solutions of this equation will be the two numbers we are looking for. We can check that they indeed multiply to \(a\) and add to \(b\).

Step 6 ：The final answer is \(\boxed{\left(\frac{b + \sqrt{b^2 - 4a}}{2}, \frac{b - \sqrt{b^2 - 4a}}{2}\right)}\).