To solve the equation \( x^2 - 2x = 24 \) by completing the square, follow these steps:
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Move the constant term to the right side of the equation: \[ x^2 - 2x - 24 = 0 \]
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Take the \( x^2 - 2x \) part and complete the square:
- To complete the square, take half of the coefficient of \( x \) (which is \(-2\)), square it, and add to both sides. Half of \(-2\) is \(-1\), and squaring it gives \(1\).
- Add \(1\) to both sides: \[ x^2 - 2x + 1 = 24 + 1 \] This simplifies to: \[ (x - 1)^2 = 25 \]
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Solve for \( x \) by taking the square root of both sides: \[ x - 1 = \pm 5 \]
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Now solve for \( x \):
- For the positive case: \[ x - 1 = 5 \implies x = 6 \]
- For the negative case: \[ x - 1 = -5 \implies x = -4 \]
Thus, the solutions to the equation \( x^2 - 2x = 24 \) are: \[ \boxed{6 \text{ and } -4} \]