To factor the polynomial \(7x^2 + 42x + 63 = 0\), we can start by factoring out the greatest common factor (GCF) from all the terms.
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Identify the GCF of the coefficients: The coefficients are \(7\), \(42\), and \(63\), and the GCF is \(7\).
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Factor out \(7\):
\[ 7(x^2 + 6x + 9) = 0 \]
- Now, we can factor the quadratic \(x^2 + 6x + 9\). We need two numbers that multiply to \(9\) (the constant term) and add up to \(6\) (the linear coefficient).
The two numbers that satisfy this are \(3\) and \(3\):
\[ x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2 \]
- Therefore, we can rewrite the polynomial:
\[ 7(x + 3)^2 = 0 \]
- To find the roots, we set \(7(x + 3)^2 = 0\):
\[ (x + 3)^2 = 0 \]
This gives us:
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]
Thus, the completely factored form of the original expression is:
\[ 7(x + 3)^2 = 0 \]
This indicates that the polynomial has a double root at \(x = -3\).
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