Let x represent one of the unknown numbers and y represent the other.
The product of both numbers is 24: xy=24
The sum of both numbers is 14: x+y=24
With this we determined a 2 unknown equation system.
Now write one of the equation is terms of one of the variables, for example write the second equation for x:

Replace it in the first one

Solve the parentheses applying the distributive propperty of multiplication:

Set the equal to zero:

Using the quadratic formula solve for the possible values of y:
![y=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}](https://img.qammunity.org/2023/formulas/mathematics/college/gfo69urp8rw17yg122uzk6md9epfe1uf75.png)
For the expression determined a=-1, b=14 and c=-24, replace in the formula and calculate
![\begin{gathered} y=\frac{-14\pm\sqrt[]{14^2-4(-1)(-24)}}{2(-1)} \\ y=\frac{-14\pm\sqrt[]{100}}{-2} \\ y=(-14\pm10)/(-2) \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/f1x8ljeaz2z9m62iltp5y0b90hw7i44va6.png)
Now solve for the two possible values of y
Positive:

Negative:

y has two possible outcomes 2 and 12, for both values you have to calculate the value of x using either equation:
For y=2

For y=12

The numbers whose product is 24 and sum is 14 are 2 and 12