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

.(1 point)

1 answer

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

  1. Move the constant to the other side of the equation:

\[ x^2 - 22x + 1 - 15 = 0 \] \[ x^2 - 22x - 14 = 0 \]

  1. Rearrange the equation for completion of the square:

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

  1. Find the coefficient of \( x \), divide by 2, and square it:

The coefficient of \( x \) is \(-22\). Half of \(-22\) is \(-11\), and squaring it gives:

\[ (-11)^2 = 121 \]

  1. Add and subtract this square inside the equation:

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

  1. Now, move 121 to the other side:

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

  1. Take the square root of both sides:

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

This can be simplified since \( \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15} \):

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

  1. Solve for \( x \):

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

Thus, the solutions are:

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