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:
- \(x \cdot y = 30\)
- \(x + y = -13\)
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:
- \(x + 3 = 0 \implies x = -3\)
- \(x + 10 = 0 \implies x = -10\)
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 \(-10\).
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