To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, follow these steps:
-
Start by moving all terms to one side of the equation: \[ x^2 - 22x + 1 - 15 = 0 \] This simplifies to: \[ x^2 - 22x - 14 = 0 \]
-
Next, isolate the \( x^2 \) and \( x \) terms: \[ x^2 - 22x = 14 \]
-
Complete the square on the left side. Take the coefficient of \( x \), which is \(-22\), divide it by 2 to get \(-11\), and then square it: \[ (-11)^2 = 121 \]
-
Add \( 121 \) to both sides of the equation: \[ x^2 - 22x + 121 = 14 + 121 \] Simplifying the right side gives: \[ x^2 - 22x + 121 = 135 \]
-
The left side can now be factored as a perfect square: \[ (x - 11)^2 = 135 \]
-
Take the square root of both sides: \[ x - 11 = \pm \sqrt{135} \]
-
Solve for \( x \): \[ x = 11 \pm \sqrt{135} \]
Thus, the solutions are: \[ x = 11 + \sqrt{135}, \quad x = 11 - \sqrt{135} \]
The numerical simplification of \( \sqrt{135} \) is \( 3\sqrt{15} \), so we can also express the solutions as: \[ x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15} \]
Based on the options given, the correct response is: x equals 11 plus 3√15, 11 minus 3√15.