Question

Write two numbers that multiply to the value on top and add to the value on bottom

30

-13

Answer 1

To find two numbers that multiply to 30 and add up to -13, we can denote the two numbers as \(x\) and \(y\). We need to solve the following equations:

1. \(x \cdot y = 30\)
2. \(x + y = -13\)

We can express \(y\) in terms of \(x\) using the second equation:

\[ y = -13 - x \]

Now, substitute this expression for \(y\) into the first equation:

\[ x \cdot (-13 - x) = 30 \]

Expanding this gives:

\[ -13x - x^2 = 30 \]

Rearranging it into standard form:

\[ x^2 + 13x + 30 = 0 \]

Now, we can factor this quadratic:

\[ (x + 3)(x + 10) = 0 \]

Setting each factor to zero gives the solutions:

1. \(x + 3 = 0 \implies x = -3\)
2. \(x + 10 = 0 \implies x = -10\)

So, the two numbers are \(-3\) and \(-10\).

Verification:
- Multiply: \(-3 \cdot -10 = 30\)
- Add: \(-3 + (-10) = -13\)

Thus, the two numbers are \(-3\) and \(-10\).