Question

im thinking of 2 numbers. The sum of my number is 55. The product of my numbers is 684. What are the 2 numbers? Explain.

Answer 1

Let x and y be the two number. Then, we can write

$\begin{gathered} x+y=55 \\ x\cdot y=684 \end{gathered}$

From the first equation, we have

$y=55-x$

By substituting this result into the second equation, we obtain

$x\mathrm{}(55-x)=684$

which gives

$55x-x^2=684$

we can rewrite this quadratic equation as follows

$x^2-55x+684=0$

Then, we can apply the quadratic formula, that is,

$x=\frac{-(-55)\pm\sqrt[]{(-55)^2-4(1)(684)}}{2}$

which gives

$\begin{gathered} x=\frac{55+\sqrt[]{3025-2736}}{2} \\ x=\frac{55\pm17}{2} \end{gathered}$

Then, the 2 solutions for x are

$\begin{gathered} x=\frac{72}{2}=36 \\ x=\frac{38}{2}=19 \end{gathered}$

Now, we can substitute these solutions into the equation x.y=684. For the first solution, we have

$\begin{gathered} 36\cdot y=684\Rightarrow y=\frac{684}{36}=19 \\ \end{gathered}$

and for the second solution, we have

$19\cdot y=684\Rightarrow y=\frac{684}{19}=36$

Therefore, the two numbers are 19 and 36