To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, follow these steps:
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Rearrange the equation: Move 15 to the left side: \[ x^2 - 22x + 1 - 15 = 0 \] This simplifies to: \[ x^2 - 22x - 14 = 0 \]
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Isolate the quadratic and linear terms: \[ x^2 - 22x = 14 \]
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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.
- Squaring -11 gives 121.
- Add 121 to both sides: \[ x^2 - 22x + 121 = 14 + 121 \] This simplifies to: \[ (x - 11)^2 = 135 \]
- Take half of the coefficient of \(x\) (which is -22), square it, and add it to both sides:
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Take the square root of both sides: \[ x - 11 = \pm \sqrt{135} \]
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Solve for \(x\): \[ x = 11 \pm \sqrt{135} \]
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Simplify \(\sqrt{135}\): \[ \sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15} \] Thus, the solutions can be written as: \[ x = 11 \pm 3\sqrt{15} \]
So the final solutions to the equation \( x^2 - 22x + 1 = 15 \) are: \[ x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15} \]