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

Let the first number be \(x\)
Thus the second number can be represented as \(27 - x\).

Since the product of the two numbers is 182,

\(\Rightarrow x(27 - x) = 182\)
\(\Rightarrow 27x - x^2 - 182 = 0\)
\(\Rightarrow x^2 - 27x + 182 = 0\)
(Splitting -27x as -13x - 14x )
\(\Rightarrow x^2 - 13x - 14x + 182 = 0\)
\(\Rightarrow x(x - 13) - 14(x - 13) = 0\)
\(\Rightarrow (x - 14)(x - 13) = 0\)

The roots of this equation are the values of x for which \( (x - 14)(x - 13) = 0 \)
which are,
\( x - 14 = 0 \) or \( x - 13 = 0\)
Thus, \(x = 14\) or \(x = 13\)

If the first number is 14, the second number will be 27 - 14 = 13.

If the first number is 13, the second number will be 27 -13 = 14.

Thus, the two numbers are 13 and 14.