To factor the polynomial expression \( 7x^2 + 42x + 63 = 0 \), we can start by factoring out the greatest common factor (GCF) from the polynomial. The GCF of the coefficients \( 7 \), \( 42 \), and \( 63 \) is \( 7 \).
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Factor out \( 7 \): \[ 7(x^2 + 6x + 9) = 0 \]
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Next, we need to factor the quadratic expression \( x^2 + 6x + 9 \). We can do this by looking for two numbers that multiply to \( 9 \) (the constant term) and add up to \( 6 \) (the coefficient of \( x \)). The numbers \( 3 \) and \( 3 \) work: \[ x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2 \]
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Putting it all together, we have: \[ 7(x + 3)^2 = 0 \]
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To find the roots of the equation, set the factored expression equal to zero: \[ 7(x + 3)^2 = 0 \]
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Since \( 7 \neq 0 \), we can divide both sides by \( 7 \): \[ (x + 3)^2 = 0 \]
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Now, take the square root of both sides: \[ x + 3 = 0 \]
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Solving for \( x \): \[ x = -3 \]
Thus, the solution to the equation \( 7x^2 + 42x + 63 = 0 \) is: \[ x = -3 \]
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