To solve the quadratic equation \(x^2 + 8x = 9\) by completing the square, follow these steps:
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Move the constant to the other side: \[ x^2 + 8x - 9 = 0 \]
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Complete the square:
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Take the coefficient of \(x\) (which is 8), divide it by 2, and square it: \[ \left(\frac{8}{2}\right)^2 = 4^2 = 16 \]
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Add and subtract this value inside the equation: \[ x^2 + 8x + 16 - 16 - 9 = 0 \]
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Rearranging gives: \[ (x + 4)^2 - 25 = 0 \]
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Isolate the squared term: \[ (x + 4)^2 = 25 \]
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Take the square root of both sides: \[ x + 4 = \pm 5 \]
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Solve for \(x\):
- For the positive case: \[ x + 4 = 5 \implies x = 5 - 4 = 1 \]
- For the negative case: \[ x + 4 = -5 \implies x = -5 - 4 = -9 \]
Thus, the two values that solve the equation \(x^2 + 8x = 9\) are: \[ \boxed{1} \quad \text{and} \quad \boxed{-9} \]