omplete the square to solve the following quadratic equation: x2−22x+1=15

.(1 point)
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To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, follow these steps:

  1. Rearrange the equation: Move 15 to the left side: \[ x^2 - 22x + 1 - 15 = 0 \] This simplifies to: \[ x^2 - 22x - 14 = 0 \]

  2. Isolate the quadratic and linear terms: \[ x^2 - 22x = 14 \]

  3. 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: \[ (x - 11)^2 = 135 \]
  4. Take the square root of both sides: \[ x - 11 = \pm \sqrt{135} \]

  5. Solve for \(x\): \[ x = 11 \pm \sqrt{135} \]

  6. 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: \[ x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15} \]