Question
Sophia was solving the following quadratic equation by factoring: `x^{2}-8x-20=0`
what is the answer
what is the answer
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Answered by
GPT-4o mini
To solve the quadratic equation \(x^{2} - 8x - 20 = 0\) by factoring, we need to find two numbers that multiply to \(-20\) (the constant term) and add to \(-8\) (the coefficient of the linear term).
The numbers that meet these criteria are \(-10\) and \(2\) because:
- \( -10 \times 2 = -20 \)
- \( -10 + 2 = -8 \)
Using these numbers, we can factor the quadratic as follows:
\[
x^{2} - 8x - 20 = (x - 10)(x + 2) = 0
\]
Now, we set each factor equal to zero:
1. \(x - 10 = 0\) → \(x = 10\)
2. \(x + 2 = 0\) → \(x = -2\)
Thus, the solutions to the equation \(x^{2} - 8x - 20 = 0\) are:
\[
\boxed{10} \quad \text{and} \quad \boxed{-2}
\]
The numbers that meet these criteria are \(-10\) and \(2\) because:
- \( -10 \times 2 = -20 \)
- \( -10 + 2 = -8 \)
Using these numbers, we can factor the quadratic as follows:
\[
x^{2} - 8x - 20 = (x - 10)(x + 2) = 0
\]
Now, we set each factor equal to zero:
1. \(x - 10 = 0\) → \(x = 10\)
2. \(x + 2 = 0\) → \(x = -2\)
Thus, the solutions to the equation \(x^{2} - 8x - 20 = 0\) are:
\[
\boxed{10} \quad \text{and} \quad \boxed{-2}
\]
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