To complete the square for the quadratic equation \( x^2 + 8x = 9 \), 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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Add \(\left(\frac{b}{2}\right)^2\) to both sides, where \(b\) is the coefficient of \(x\). In this case, \(b = 8\): \[ \left(\frac{8}{2}\right)^2 = 4^2 = 16 \]
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Add 16 to both sides: \[ x^2 + 8x + 16 = 9 + 16 \] \[ x^2 + 8x + 16 = 25 \]
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Now, rewrite the left side as a binomial squared: \[ (x + 4)^2 = 25 \]
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Take the square root of both sides: \[ x + 4 = \pm 5 \]
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Solve for \(x\): \[ x + 4 = 5 \quad \Rightarrow \quad x = 1 \] \[ x + 4 = -5 \quad \Rightarrow \quad x = -9 \]
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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