To complete the square for the quadratic equation \( x^2 + 8x = 9 \), we follow these steps:
- Move the constant to the other side of the equation:
\[ x^2 + 8x - 9 = 0 \]
- Rearrange it slightly:
\[ x^2 + 8x = 9 \]
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To complete the square, we need to 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 \( 4^2 = 16 \).
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Now add 16 to both sides of the equation:
\[ x^2 + 8x + 16 = 9 + 16 \]
\[ x^2 + 8x + 16 = 25 \]
- Now, the left side can be factored as a perfect square:
\[ (x + 4)^2 = 25 \]
- To solve for \( x \), take the square root of both sides:
\[ x + 4 = \pm 5 \]
- Now, solve for \( x \) in both cases:
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For the positive case: \[ x + 4 = 5 \implies x = 5 - 4 \implies x = 1 \]
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For the negative case: \[ x + 4 = -5 \implies x = -5 - 4 \implies x = -9 \]
Therefore, the two solutions to the equation \( x^2 + 8x = 9 \) are
\[ \boxed{1} \text{ and } \boxed{-9}. \]