Let's call x and y as the required numbers.
Their sum is 42, thus:
x + y = 42
Their product is 432, thus:
xy = 432
From the first equation, solve for x:
x = 42 - y
Substitute in the second equation:
(42 - y) y = 432
Multiplying:
[tex]42y-y^2=432[/tex]Rearranging:
[tex]-y^2+42y-432=0[/tex]We use the quadratic formula to find the solutions to this equation. For a=-1, b=42, and c=-432:
[tex]y=\frac{-42\pm\sqrt[]{42^2-4\cdot(-1)\cdot(-432)}}{2\cdot(-1)}[/tex]Calculating:
[tex]\begin{gathered} y=\frac{-42\pm\sqrt[]{1764-1728}}{-2} \\ y=\frac{-42\pm\sqrt[]{36}}{-2} \\ y=\frac{-42\pm6}{-2} \end{gathered}[/tex]Separate the two roots:
y = 24 , y = 18
If we use the first value, then x = 42 - 24 = 18
If we use the second value, then x = 42 - 18 = 24
In either case, the two integers are 18 and 24
