To solve the equation \( 3p^2 - 18p + 24 = 0 \) by factoring, we'll first factor the quadratic expression.
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Start with the original equation: \[ 3p^2 - 18p + 24 = 0 \]
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Factor out the greatest common factor (GCF) from all terms: \[ 3(p^2 - 6p + 8) = 0 \]
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Now, we will factor the quadratic \( p^2 - 6p + 8 \). We need two numbers that multiply to \( 8 \) (the constant term) and add up to \( -6 \) (the coefficient of \( p \)). The numbers \( -2 \) and \( -4 \) fit these criteria.
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Thus, we can factor the quadratic as: \[ 3(p - 2)(p - 4) = 0 \]
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Now we can apply the zero product property. If the product of two factors equals zero, at least one of the factors must be zero: \[ 3 \neq 0 \quad \text{(so we can ignore this)} \] \[ p - 2 = 0 \quad \text{or} \quad p - 4 = 0 \]
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Solving for \( p \) gives: \[ p - 2 = 0 \quad \Rightarrow \quad p = 2 \] \[ p - 4 = 0 \quad \Rightarrow \quad p = 4 \]
So, the solutions to the equation \( 3p^2 - 18p + 24 = 0 \) are:
\[ p = 2, 4 \]
Thus, the correct answer is:
p = 2, 4.