Question

Find the 2 numbers. Write your answer like this: 4,9 The product of two positive numbers is 65. One number is 3 more than two times the smaller number.

Answer 1

Let the smaller number be x

Then,

$\begin{gathered} 2x+3=y \\ \text{and} \\ xy=65 \end{gathered}$

Substitute 2x+3 for y in xy=65.

$\begin{gathered} x(2x+3)=65 \\ 2x^2+3x=65 \end{gathered}$

So this forms a qudratic equation,

$\begin{gathered} x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ \text{ =}\frac{\text{-3}\pm\sqrt{9-4\times2\times(-65)}}{2\times2} \\ \text{ =-3}\pm\frac{\sqrt{529}}{4} \\ \text{ =}\frac{\text{-3}\pm23}{4} \\ =5 \end{gathered}$

So the smallest number is 5. And

$y=2\times5+3=13$

The answer is (13,5)