To solve the equation \( x^2 + 14x + 45 = 0 \) by completing the square, follow these steps:
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Move the constant to the other side of the equation:
\[ x^2 + 14x = -45 \]
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Complete the square for the left side:
Take the coefficient of \( x \), which is \( 14 \), divide it by \( 2 \), and square it:
\[ \left(\frac{14}{2}\right)^2 = 7^2 = 49 \]
Add \( 49 \) to both sides of the equation:
\[ x^2 + 14x + 49 = -45 + 49 \]
Simplifying the right side:
\[ x^2 + 14x + 49 = 4 \]
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Rewrite the left side as a squared term:
The left side can now be factored:
\[ (x + 7)^2 = 4 \]
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Take the square root of both sides:
\[ x + 7 = \pm 2 \]
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Solve for \( x \):
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If \( x + 7 = 2 \):
\[ x = 2 - 7 = -5 \]
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If \( x + 7 = -2 \):
\[ x = -2 - 7 = -9 \]
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Final answers:
The solutions to the equation \( x^2 + 14x + 45 = 0 \) are:
\[ x = -5 \quad \text{and} \quad x = -9 \]
Thus, the values of \( x \) are \( \boxed{-5} \) and \( \boxed{-9} \).
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