To factor the polynomial expression \(7x^2 + 42x + 63 = 0\), we can start by factoring out the greatest common factor (GCF) from the entire polynomial.
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Identify the GCF: The GCF of the coefficients (7, 42, and 63) is 7.
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Factor out the GCF: \[ 7(x^2 + 6x + 9) = 0 \]
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Factor the quadratic: Next, we can factor the quadratic expression \(x^2 + 6x + 9\). This can be factored as: \[ (x + 3)(x + 3) \quad \text{or} \quad (x + 3)^2 \]
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Write the complete factorization: \[ 7(x + 3)^2 = 0 \]
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Set the factors to zero: Now we can solve for \(x\) by setting each factor equal to zero: \[ 7 = 0 \quad (\text{no solution here}), \] and \[ (x + 3)^2 = 0 \]
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Solve for \(x\): \[ x + 3 = 0 \] \[ x = -3 \]
Therefore, the solution to the equation \(7x^2 + 42x + 63 = 0\) is: \[ \boxed{-3} \]
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