Question

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

Answer 1

Hint: We will form two equations with two unknowns using the given information. We will obtain a quadratic equation in one variable by substitution. Then we will solve the quadratic equation using the quadratic formula. The solution of the quadratic formula will be one of the two required numbers. Using the solution of the quadratic equation, we will be able to find the other number.

Complete step by step answer:
Let the two numbers be $x$ and $y$. According to the given information, we have the following two equations,
$x+y=27$....(i)
$x\times y=182$....(ii)
From equation (i), we can write $y=27-x$. Substituting this value in equation (ii), we get
$\begin{align}
  & x\times \left( 27-x \right)=182 \\
 & \therefore 27x-{{x}^{2}}=182 \\
\end{align}$
Rearranging the above equation, we get the following quadratic equation,
${{x}^{2}}-27x+182=0$
The general quadratic equation is $a{{x}^{2}}+bx+c=0$ and the quadratic formula for solving this equation is $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$. We will compare the above quadratic equation with the general quadratic equation, so we have $a=1,\text{ }b=-27,\text{ }c=182$. We will substitute these values in the quadratic formula to find the values of $x$ in the following manner,
$x=\dfrac{-\left( -27 \right)\pm \sqrt{{{27}^{2}}-4\times 1\times 182}}{2\times 1}$
Simplifying the above equation, we get
$\begin{align}
  & x=\dfrac{27\pm \sqrt{729-728}}{2} \\
 &\Rightarrow x =\dfrac{27\pm 1}{2}
\end{align}$
Therefore, we get $x=\dfrac{28}{2}=14$ or $x=\dfrac{26}{2}=13$.
Now, substituting these values in equation (i), we get
for $x=14$, we have $y=27-14=13$ and for $x=13$, we have $y=27-13=14$

Therefore, the two numbers whose sum is 27 and product is 182 are 13 and 14.

Note: There is another method to solve this question which involves solving a quadratic equation with a different method. If we are familiar with the method of solving a quadratic equation by factorization, then we can reach the answer in fewer steps. The substitutions need to be done explicitly to avoid confusion.