To solve the equation \( x^2 + 12x = -20 \) by completing the square, follow these steps:
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Move the constant term to the other side: \[ x^2 + 12x + 20 = 0 \] (We can actually just keep the equation \( x^2 + 12x = -20 \) and add 20 to both sides later, but moving it here clarifies the next step.)
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Identify \( b \): In the equation \( x^2 + 12x \), \( b = 12 \).
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Calculate \( \left( \frac{b}{2} \right)^2 \): \[ \left( \frac{12}{2} \right)^2 = 6^2 = 36 \]
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Add \( \left( \frac{b}{2} \right)^2 \) to both sides of the equation: \[ x^2 + 12x + 36 = -20 + 36 \] \[ x^2 + 12x + 36 = 16 \]
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Write the left side as a binomial squared: \[ (x + 6)^2 = 16 \]
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Use the square root property of equality: \[ x + 6 = \pm 4 \]
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Isolate the variable:
- For \( x + 6 = 4 \): \[ x = 4 - 6 = -2 \]
- For \( x + 6 = -4 \): \[ x = -4 - 6 = -10 \]
Thus, the solutions to the equation \( x^2 + 12x = -20 \) are \( x = -2 \) and \( x = -10 \).