To solve the quadratic equation \(x^2 + 8x = 9\) by completing the square, follow these steps:
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Rewrite the equation: Move \(9\) to the left side of the equation: \[ x^2 + 8x - 9 = 0 \]
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Rearrange: Isolate the \(x^2 + 8x\) term: \[ x^2 + 8x = 9 \]
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Complete the square: 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 square on the left side: \[ x^2 + 8x + 16 - 16 = 9 \] This can be rewritten as: \[ (x + 4)^2 - 16 = 9 \]
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Simplify: Now add \(16\) to both sides: \[ (x + 4)^2 = 9 + 16 \] \[ (x + 4)^2 = 25 \]
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Take the square root of both sides: Remember to consider both the positive and negative roots: \[ x + 4 = \pm 5 \]
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Solve for \(x\): This gives us two equations to solve:
- 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 solutions to the equation \(x^2 + 8x = 9\) are: \[ \boxed{1 \text{ and } -9} \]