To complete the square for the quadratic equation \( x^2 + 8x = 9 \), we will follow these steps:
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Move the constant term from the right side to the left side: \[ x^2 + 8x - 9 = 0 \]
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Isolate the \(x\) terms: \[ x^2 + 8x = 9 \]
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To complete the square, take the coefficient of \(x\) (which is 8), divide it by 2 (resulting in 4), and then square it (resulting in 16). Add this square to both sides of the equation: \[ x^2 + 8x + 16 = 9 + 16 \] Simplifying this gives: \[ x^2 + 8x + 16 = 25 \]
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Now, the left side of the equation can be factored: \[ (x + 4)^2 = 25 \]
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To find \(x\), take the square root of both sides: \[ x + 4 = \pm 5 \]
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This gives us two equations to solve: \[ x + 4 = 5 \quad \text{or} \quad x + 4 = -5 \]
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Solving these:
- For \(x + 4 = 5\): \[ x = 5 - 4 = 1 \]
- For \(x + 4 = -5\): \[ x = -5 - 4 = -9 \]
Thus, the two values that solve the equation \(x^2 + 8x = 9\) are: \[ \boxed{1} \quad \text{and} \quad \boxed{-9} \]