To solve the quadratic equation \( x^2 + 8x = 9 \) by completing the square, we first rearrange the equation by moving 9 to the left side:
\[ x^2 + 8x - 9 = 0 \]
Now, to complete the square, we focus on the quadratic and linear terms \( x^2 + 8x \).
- Take the coefficient of \( x \) (which is 8), divide it by 2, and square it:
\[ \left(\frac{8}{2}\right)^2 = 4^2 = 16 \]
- Add and subtract this value (16) on the left side:
\[ x^2 + 8x + 16 - 16 - 9 = 0 \]
This simplifies to:
\[ (x + 4)^2 - 25 = 0 \]
- Now, move -25 to the right side:
\[ (x + 4)^2 = 25 \]
- Taking the square root of both sides, we have:
\[ x + 4 = 5 \quad \text{or} \quad x + 4 = -5 \]
- Solving for \( x \):
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From \( x + 4 = 5 \): \[ x = 5 - 4 = 1 \]
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From \( x + 4 = -5 \): \[ x = -5 - 4 = -9 \]
Thus, the two values that solve the equation are:
\[ \boxed{1} \quad \text{and} \quad \boxed{-9} \]
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