Complete the square to solve the following quadratic equation: x^2 - 22x + 1 = 15.

A: x = -11 + 3 to the square root of 15, -11 -3 to the square root of 15.

B: x = 11 + 3 to the square root of 15, 11 - 3 to the square root of 15

C: x = -11 + to the square root of 14, -11 - to the square root of 14.

D: x = 11 + to the square root of 14, 11 - to the square root of 14

1 answer

To solve the quadratic equation \( x^2 - 22x + 1 = 15 \) by completing the square, we first rearrange the equation:

\[ x^2 - 22x + 1 - 15 = 0 \]

This simplifies to:

\[ x^2 - 22x - 14 = 0 \]

Next, we will complete the square for the quadratic expression \( x^2 - 22x \).

  1. Take half of the coefficient of \( x \) (which is -22), and square it: \[ \left(-\frac{22}{2}\right)^2 = (-11)^2 = 121 \]

  2. Add and subtract 121 inside the equation:

\[ x^2 - 22x + 121 - 121 - 14 = 0 \] \[ (x - 11)^2 - 135 = 0 \]

Now, we'll move -135 to the other side:

\[ (x - 11)^2 = 135 \]

Next, we'll take the square root of both sides:

\[ x - 11 = \pm \sqrt{135} \]

Now, solving for \( x \):

\[ x = 11 \pm \sqrt{135} \]

Next, we simplify \( \sqrt{135} \):

\[ \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15} \]

This gives us:

\[ x = 11 \pm 3\sqrt{15} \]

So the solutions are:

\[ x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15} \]

Now we can match these with the provided options.

Looking through the choices:

  • A: \( x = -11 + 3\sqrt{15}, -11 - 3\sqrt{15} \)
  • B: \( x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15} \)
  • C: \( x = -11 + \sqrt{14}, -11 - \sqrt{14} \)
  • D: \( x = 11 + \sqrt{14}, 11 - \sqrt{14} \)

The correct answer is:

B: \( x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15} \).