To solve the quadratic equation \( x^2 + 8x = 9 \) by completing the square, we first rearrange the equation to set it to zero:
\[ x^2 + 8x - 9 = 0 \]
Next, we need to complete the square for the expression \( x^2 + 8x \). To do this, we take the coefficient of \( x \), which is 8, divide it by 2, and then square it:
\[ \left(\frac{8}{2}\right)^2 = 4^2 = 16 \]
Now we can rewrite \( x^2 + 8x \) as a perfect square. We add and subtract 16 in the equation:
\[ x^2 + 8x + 16 - 16 - 9 = 0 \]
This simplifies to:
\[ (x + 4)^2 - 25 = 0 \]
Now we can isolate the perfect square:
\[ (x + 4)^2 = 25 \]
Next, we take the square root of both sides:
\[ x + 4 = \pm 5 \]
This leads to two equations:
- \( x + 4 = 5 \)
- \( x + 4 = -5 \)
Solving these equations gives us:
- \( x = 5 - 4 \) which simplifies to \( x = 1 \)
- \( x = -5 - 4 \) which simplifies to \( x = -9 \)
Thus, the two values that solve the quadratic equation \( x^2 + 8x = 9 \) are:
\[ \boxed{1} \quad \text{and} \quad \boxed{-9} \]
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