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)=(-24\pm√(24^2-4\cdot\:1\cdot\left(-756\right)))/(2\cdot\:1) \\ y_(1,\: 2)=(-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