To solve the equation \( v^2 - 8v - 20 = 0 \) by factoring, we first need to find two numbers that multiply to \(-20\) (the constant term) and add up to \(-8\) (the coefficient of the \(v\) term).
The two numbers that work are \(-10\) and \(2\) because:
\[ -10 \times 2 = -20 \quad \text{(product)} \] \[ -10 + 2 = -8 \quad \text{(sum)} \]
Now we can factor the quadratic equation:
\[ v^2 - 8v - 20 = (v - 10)(v + 2) = 0 \]
Next, we set each factor to zero:
- \( v - 10 = 0 \) → \( v = 10 \)
- \( v + 2 = 0 \) → \( v = -2 \)
Thus, the solutions to the equation are:
\[ v = 10 \quad \text{and} \quad v = -2 \]
So, the final answer is:
- The solutions are \( v = 10 \) and \( v = -2 \).
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