To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, follow these steps:
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Move all terms to one side of the equation: \[ x^2 - 22x + 1 - 15 = 0 \] This simplifies to: \[ x^2 - 22x - 14 = 0 \]
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Isolate the constant term on one side: \[ x^2 - 22x = 14 \]
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Complete the square: To complete the square, take half of the coefficient of \( x \) (which is \( -22 \)), square it, and add it to both sides. Half of \( -22 \) is \( -11 \), and squaring it gives \( 121 \).
So, add \( 121 \) to both sides: \[ x^2 - 22x + 121 = 14 + 121 \] This simplifies to: \[ x^2 - 22x + 121 = 135 \]
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Express the left-hand side as a squared term: The left side can be factored as a perfect square: \[ (x - 11)^2 = 135 \]
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Take the square root of both sides: \[ x - 11 = \pm \sqrt{135} \]
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Solve for \( x \): \[ x = 11 \pm \sqrt{135} \]
Since \( \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15} \), we can rewrite the solutions as: \[ x = 11 \pm 3\sqrt{15} \]
The final solutions are: \[ x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15} \]