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

Question

Homi · Accepted Answer

Understand the Question

Let one of the numbers be $x$ . Since their sum is $27$ , the other number is $27 - x$ .
Their product is given as $182$ , which allows us to set up a quadratic equation: $x(27 - x) = 182$ .
Solving this quadratic equation by factorisation gives the values of the two numbers.

Step 1 · Formulate the Quadratic Equation

Let the first number be $x$ . Since the sum of the two numbers is $27$ , the second number is $27 - x$ .

Given their product is $182$

\begin{aligned} x(27 - x) &= 182 \\ 27x - x^2 &= 182 \\ x^2 - 27x + 182 &= 0 \end{aligned}

Step 2 · Solve by Factorisation

Find two numbers whose sum is $-27$ and product is $182$ . The numbers are $-13$ and $-14$ .

\begin{aligned} x^2 - 13x - 14x + 182 &= 0 \\ x(x - 13) - 14(x - 13) &= 0 \\ (x - 13)(x - 14) &= 0 \end{aligned}

Step 3 · Find the Numbers

Setting each factor to zero:

\begin{aligned} x - 13 &= 0 \implies x = 13 \\ x - 14 &= 0 \implies x = 14 \end{aligned}

Answer

$13$ and $14$

Common Mistakes

Sign Error in Splitting Middle Term: Selecting $+13$ and $+14$ gives a sum of $+27$ , whereas the coefficient is $-27$ . Both factors must be negative ( $-13$ and $-14$ ) so that their product is $+182$ and sum is $-27$ .
Incomplete Answer: Stopping after finding $x = 13$ or $x = 14$ without calculating and stating both required numbers.

Question 3