For this problem, we need to find two positive numbers whose difference is 5 and the product is 864.
We will call the numbers: x and y. So we can write the following expressions:
[tex]\begin{gathered} x-y=5\\ \\ x\cdot y=864 \end{gathered}[/tex]
We can rewrite the first expression as:
[tex]x=5+y[/tex]
Then we can replace the expression above with the second one:
[tex]\begin{gathered} (5+y)\cdot y=864\\ \\ y^2+5y-864=0\\ \\ \\ \end{gathered}[/tex]
We can solve the expression above to determine one of the numbers.
[tex]\begin{gathered} y_{1,2}=\frac{-5\pm\sqrt{(5)^2-4\cdot1\cdot(-864)}}{2}\\ \\ y_{1,2}=\frac{-5\pm\sqrt{3481}}{2}\\ \\ y_{1,2}=\frac{-5\pm59}{2}\\ \\ y_1=\frac{-5+59}{2}=27\\ \\ y_2=\frac{-5-59}{2}=-32 \end{gathered}[/tex]
The only possible value we can use is "27". We can find the other number by replacing "y" on the first expression with 27. We have:
[tex]x=5+27=32[/tex]
The numbers are 27 and 32.
