What two numbers multiply to -8 and add up to 3?

Steps to Solve

1. List factor pairs of -8

We need to find two numbers that multiply to -8. Considering both positive and negative factors, we have the following pairs: $$(1, -8), (-1, 8), (2, -4), (-2, 4)$$

2. Calculate the sum of each factor pair

Now, we add each of these pairs together to see which pair sums to 3:

$1 + (-8) = -7$ $-1 + 8 = 7$ $2 + (-4) = -2$ $-2 + 4 = 2$

We made an error in the previous step, there is one more factor pair.

3. Correct the pairs and Calculate the sum of each factor pair

$$(1, -8), (-1, 8), (2, -4), (-2, 4)$$

Now, we add each of these pairs together to see which pair sums to 3:

$1 + (-8) = -7$ $-1 + 8 = 7$ $2 + (-4) = -2$ $-2 + 4 = 2$

We need to account for the fact that 8 can be written as $222$. This means that $8$ actually has more factors than expected. The full factor pairs are

$$ (1, -8), (-1, 8), (2, -4), (-2, 4) $$

3. Re-evaluate factor pairs and check their sums

Upon closer inspection, we realize that none of the integer pairs multiply to -8 and add up to 3. This suggests we need to consider non-integer solutions. Let's denote the two numbers as $x$ and $y$. We have the following system of equations:

$$ xy = -8 $$ $$ x + y = 3 $$

4. Solve for one variable in terms of the other

From the second equation, we can express $y$ in terms of $x$:

$$ y = 3 - x $$

5. Substitute into the first equation

Substitute this expression for $y$ into the first equation:

$$ x(3 - x) = -8 $$

6. Rearrange into a quadratic equation

Expand and rearrange the equation into a quadratic form:

$$ 3x - x^2 = -8 $$ $$ x^2 - 3x - 8 = 0 $$

7. Solve the quadratic equation

Use the quadratic formula to solve for $x$:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

where $a = 1$, $b = -3$, and $c = -8$.

$$ x = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(-8)}}{2(1)} $$ $$ x = \frac{3 \pm \sqrt{9 + 32}}{2} $$ $$ x = \frac{3 \pm \sqrt{41}}{2} $$

8. Find the two values of x

So, the two possible values for $x$ are:

$$ x_1 = \frac{3 + \sqrt{41}}{2} $$ $$ x_2 = \frac{3 - \sqrt{41}}{2} $$

9. Solve for the corresponding y values

Now, find the corresponding $y$ values using $y = 3 - x$:

$$ y_1 = 3 - \frac{3 + \sqrt{41}}{2} = \frac{6 - 3 - \sqrt{41}}{2} = \frac{3 - \sqrt{41}}{2} $$ $$ y_2 = 3 - \frac{3 - \sqrt{41}}{2} = \frac{6 - 3 + \sqrt{41}}{2} = \frac{3 + \sqrt{41}}{2} $$

10. State solution

The two numbers are $\frac{3 + \sqrt{41}}{2}$ and $\frac{3 - \sqrt{41}}{2}$.