To complete the square for the quadratic equation \( x^2 + 8x = 9 \), follow these steps:
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Move the constant term to the other side of the equation:
\[ x^2 + 8x - 9 = 0 \]
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Next, we'll complete the square on the left-hand side. 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 inside the equation:
\[ x^2 + 8x + 16 - 16 = 9 \]
Simplifying, we should move 16 to the other side:
\[ x^2 + 8x + 16 = 9 + 16 \]
\[ x^2 + 8x + 16 = 25 \]
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Now, rewrite the left-hand side as a square:
\[ (x + 4)^2 = 25 \]
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To solve for \( x \), take the square root of both sides:
\[ x + 4 = \pm 5 \]
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Solve for \( x \) by isolating it:
- For the positive:
\[ x + 4 = 5 \implies x = 5 - 4 \implies x = 1 \]
- For the negative:
\[ x + 4 = -5 \implies x = -5 - 4 \implies x = -9 \]
The solutions to the quadratic equation \( x^2 + 8x = 9 \) are:
\[ x = 1 \quad \text{and} \quad x = -9 \]