To solve the equation \( v^2 - 5v - 24 = 0 \) by factoring, we start by looking for two numbers that multiply to \(-24\) (the constant term) and add to \(-5\) (the coefficient of the linear term).
The pair of numbers that fits these criteria is \(-8\) and \(3\), because:
\[ -8 \times 3 = -24 \]
\[ -8 + 3 = -5 \]
Using these numbers, we can factor the quadratic equation:
\[ v^2 - 5v - 24 = (v - 8)(v + 3) = 0 \]
Next, we can use the zero product property, which states that if the product of two factors is zero, at least one of the factors must be zero. Thus, we set each factor to zero:
- \( v - 8 = 0 \)
- \( v + 3 = 0 \)
Now, we solve each equation:
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From \( v - 8 = 0 \): \[ v = 8 \]
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From \( v + 3 = 0 \): \[ v = -3 \]
Thus, the solutions to the equation \( v^2 - 5v - 24 = 0 \) are:
\[ v = 8 \quad \text{and} \quad v = -3 \]