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# 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.