Question

Find two positive numbers whose difference is 24 and whose product is 756.

Answer 1

We need to find two numbers that meet the following...

$\begin{gathered} x-y=24 \\ x\cdot y=756 \end{gathered}$

first, let's clear x from the first equation

$\begin{gathered} x-y+y=24+y \\ x=24+y \end{gathered}$

Second, substitute the solutions x = 24 + y into xy = 756

$\begin{gathered} xy=756 \\ (24+y)\cdot y=756 \\ 24y+y^2=756 \\ 24y+y^2-756=756-756 \\ y^2+24y-756=0 \end{gathered}$

Now, we need to solve this second grade equation

$\begin{gathered} y_{1,\: 2}=\frac{-24\pm\sqrt{24^2-4\cdot\:1\cdot\left(-756\right)}}{2\cdot\:1} \\ y_{1,\: 2}=\frac{-24\pm\:60}{2\cdot\:1} \end{gathered}$

We will obtain two values for y

$y=18,\: y=-42$

We will only use the positive value

Now, we just have to replace y=18 into x=24+y

$\begin{gathered} x=24+y \\ x=24+18 \\ x=42 \end{gathered}$

In conclusion, the two positive numbers that meet the conditions are:

18 and 42