Bot GPT 4-o mini To solve the quadratic equation \(x^2 + 8x = 9\) by completing the square, follow these steps:
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Move the constant to the other side of the equation:
\[ x^2 + 8x - 9 = 0 \]
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Isolate the quadratic and linear terms:
\[ x^2 + 8x = 9 \]
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Complete the square for the left-hand side. To do this, take half of the coefficient of \(x\) (which is 8), square it, and add it to both sides of the equation.
Half of 8 is 4, and squaring it gives:
\[ 4^2 = 16 \]
Now add 16 to both sides:
\[ x^2 + 8x + 16 = 9 + 16 \]
Simplifying the right-hand side gives:
\[ x^2 + 8x + 16 = 25 \]
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Rewrite the left-hand side as a square:
\[ (x + 4)^2 = 25 \]
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Take the square root of both sides:
\[ x + 4 = \pm 5 \]
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Solve for \(x\):
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When \(x + 4 = 5\): \[ x = 5 - 4 = 1 \]
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When \(x + 4 = -5\): \[ x = -5 - 4 = -9 \]
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Thus, the two values that solve the equation \(x^2 + 8x = 9\) are:
\[ x = 1 \quad \text{and} \quad x = -9 \]
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