Question 3

We will set up a quadratic equation. Then we will solve it to find the numbers.

Step 1 — Formulate the equation

Let's call the first number $x$. The sum of the two numbers is 27. So, the second number will be $27 - x$. Their product is given as 182. We can write this as an equation.

$x(27 - x) = 182$

$27x - x^2 = 182$

$x^2 - 27x + 182 = 0$

$\boxed{x^2 - 27x + 182 = 0}$

Step 2 — Solve the equation by factoring

We need to find two numbers. Their product is 182. Their sum is -27. These numbers are -13 and -14. Let's split the middle term.

$x^2 - 13x - 14x + 182 = 0$

$x(x - 13) - 14(x - 13) = 0$

$(x - 13)(x - 14) = 0$

$\boxed{(x - 13)(x - 14) = 0}$

Step 3 — Find the numbers

From the factored equation, we have two possibilities. Either $x - 13 = 0$ or $x - 14 = 0$.

$x - 13 = 0$

$x = 13$

If $x = 13$, the first number is 13. The second number is $27 - x$.

$27 - 13 = 14$

So the numbers are 13 and 14.

$x - 14 = 0$

$x = 14$

If $x = 14$, the first number is 14. The second number is $27 - x$.

$27 - 14 = 13$

So the numbers are 14 and 13. Both cases give the same pair of numbers.

Answer

The two numbers are 13 and 14.