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

Question

.(1 point)
Responses

x=−11+315−−√, −11−315−−√
x equals negative 11 plus 3 Start Root 15 End Root comma negative 11 minus 3 Start Root 15 End Root

x=−11+14−−√, −11−14−−√
x equals negative 11 plus Start Root 14 End Root comma negative 11 minus Start Root 14 End Root

x=11+315−−√, 11−315−−√
x equals 11 plus 3 Start Root 15 End Root comma 11 minus 3 Start Root 15 End Root

x=11+14−−√, 11−14−−√

Answer 1

To solve the quadratic equation \(x^2 - 22x + 1 = 15\) by completing the square, we start by rearranging the equation.

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

2. Now, we complete the square for the expression \(x^2 - 22x\):

   • Take half of the coefficient of \(x\) which is \(-22\) and square it, \(\left(\frac{-22}{2}\right)^2 = 121\).
   • Add and subtract this value: \[ x^2 - 22x + 121 - 121 - 14 = 0 \] This simplifies to: \[ (x - 11)^2 - 135 = 0 \]

3. Solve for \(x\): \[ (x - 11)^2 = 135 \] Take the square root of both sides: \[ x - 11 = \pm \sqrt{135} \] Thus, we have: \[ x = 11 \pm \sqrt{135} \]

4. The values can be expressed as: \[ x = 11 + \sqrt{135}, \quad x = 11 - \sqrt{135} \]

Now, \( \sqrt{135} \) can be simplified to \( 3\sqrt{15} \) (since \(135 = 9 \times 15\)), giving us: \[ x = 11 + 3\sqrt{15}, \quad x = 11 - 3\sqrt{15} \]

So the final solutions are: \[ \boxed{x = 11 + 3\sqrt{15}, 11 - 3\sqrt{15}} \]

This matches the third option in your provided responses.