To find two numbers that multiply to 182 and add to -27, let the numbers be $x$ and $y$. They must satisfy the following equations:

1. $x \cdot y = 182$
2. $x + y = -27$

Step 1: Solve for one variable

From $x + y = -27$, solve for $y$: $y = -27 - x$

Step 2: Substitute into the product equation

Substitute $y = -27 - x$ into $x \cdot y = 182$: $x \cdot (-27 - x) = 182$ $-27x - x^2 = 182$ $x^2 + 27x + 182 = 0$

Step 3: Solve the quadratic equation

The equation to solve is: $x^2 + 27x + 182 = 0$ Using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = 27$, and $c = 182$. Substitute these values: $x = \frac{-27 \pm \sqrt{27^2 - 4(1)(182)}}{2(1)}$ $x = \frac{-27 \pm \sqrt{729 - 728}}{2}$ $x = \frac{-27 \pm \sqrt{1}}{2}$ $x = \frac{-27 \pm 1}{2}$

Step 4: Find the two roots

$x = \frac{-27 + 1}{2} = \frac{-26}{2} = -13$ $x = \frac{-27 - 1}{2} = \frac{-28}{2} = -14$

Step 5: Determine $y$

If $x = -13$, then $y = -27 - (-13) = -14$.
If $x = -14$, then $y = -27 - (-14) = -13$.

Final Answer:

The two numbers are: $\boxed{-13 \text{ and } -14}$

These numbers multiply to $182$ and add to $-27$.

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Would you like more details or have additional questions?

Related Questions:

1. How does the quadratic formula work in other cases?
2. What are practical applications of this type of problem-solving?
3. Can we solve this problem graphically using a parabola?
4. How do we factorize quadratic equations by hand?
5. What is the importance of the discriminant in solving quadratics?

Tip:

Always verify your solution by substituting the values back into the original equations.