To solve the quadratic equation \( x^2 + 8x = 9 \) by completing the square, follow these steps:
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Move 9 to the left side of the equation: \[ x^2 + 8x - 9 = 0 \]
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Rearrange to isolate the quadratic and linear terms: \[ x^2 + 8x = 9 \]
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Complete the square: To complete the square for \( x^2 + 8x \), take half of the coefficient of \( x \) (which is 8), square it, and add it to both sides. Half of 8 is 4, and squaring it gives \( 4^2 = 16 \).
Add 16 to both sides: \[ x^2 + 8x + 16 = 9 + 16 \] Simplifying gives: \[ x^2 + 8x + 16 = 25 \]
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Factor the left side: The left side can be rewritten as a 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 \( x \): Now, we have two cases to solve:
- Case 1: \( x + 4 = 5 \) \[ x = 5 - 4 = 1 \]
- Case 2: \( x + 4 = -5 \) \[ x = -5 - 4 = -9 \]
Thus, the two values that solve the equation \( x^2 + 8x = 9 \) are: \[ \boxed{1} \quad \text{and} \quad \boxed{-9} \]