To solve the equation \( x^2 + 8x = 9 \) by completing the square, follow these steps:
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Move the constant to the right side of the equation: \[ x^2 + 8x - 9 = 0 \]
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Rearrange it to the form suitable for completing the square: \[ x^2 + 8x = 9 \]
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To complete the square, take the coefficient of \( x \) (which is 8), divide it by 2, and then square the result: \[ \left(\frac{8}{2}\right)^2 = 4^2 = 16 \]
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Add and subtract this square (16) to the left side of the equation: \[ x^2 + 8x + 16 - 16 = 9 \] This can be rewritten as: \[ (x + 4)^2 - 16 = 9 \]
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Now, isolate the squared term: \[ (x + 4)^2 = 9 + 16 \] \[ (x + 4)^2 = 25 \]
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Take the square root of both sides: \[ x + 4 = 5 \quad \text{or} \quad x + 4 = -5 \]
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Solve for \( x \) in each case: \[ x + 4 = 5 \implies x = 5 - 4 = 1 \] \[ x + 4 = -5 \implies x = -5 - 4 = -9 \]
Therefore, the two solutions to the equation \( x^2 + 8x = 9 \) are: \[ x = 1 \quad \text{and} \quad x = -9 \]
Final answer: \[ X = 1, -9 \]