Question

Find two numbers whose sum is $27$ and the product is $182$.

Answer 1

Determine the required numbers with given conditions

Let the two numbers be $x$ and $y$.

Given, sum is $27$ and product is $182$.

So, $x+y=27$ and $xy=182$

Here, $x=\frac{182}{y}$

So, replacing this in $x+y=27$, we get,

$\begin{array}{cc}\Rightarrow & \frac{182}{y}+y=27\\ \Rightarrow & 182+y^2=27y\\ \Rightarrow & y^2-27y+182=0\\ \Rightarrow & y^2-13y-14y+182=0\\ \Rightarrow & y(y-13)-14(y-13)=0\\ \Rightarrow & (y-13)(y-14)=0\\ \Rightarrow & y=13,14\end{array}$

So, $y$ can either be $13$ or $14$.

$\Rightarrow$ If $y=13$, then $x=27-13=14$

$\Rightarrow$ If $y=14$, then $x=27-14=13$

Therefore, the two numbers are $13$ and $14$.