Complete the square to solve the following quadratic equation: x2−22x+1=15 .(1 point) Responses 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+14−−√, 11−14−−√ x equals 11 plus Start Root 14 End Root comma 11 minus Start Root 14 End Root 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+315−−√, 11−315−−√
To complete the square, we can start by moving the constant term to the right side of the equation:
x^2 - 22x + 1 = 15
x^2 - 22x - 14 = 0
Next, take half of the coefficient of x, square it, and add it to both sides of the equation:
x^2 - 22x + (-22/2)^2 = 14 + (-22/2)^2
x^2 - 22x + 121 = 14 + 121
x^2 - 22x + 121 = 135
Now, write the left side of the equation as a perfect square binomial:
(x - 11)^2 = 135
Take the square root of both sides:
x - 11 = ±√135
Simplify the square root of 135:
x - 11 = ±√(9 × 15)
x - 11 = ±√9 × √15
x - 11 = ±3√15
Finally, solve for x:
x = 11 ± 3√15
Therefore, the solution to the quadratic equation x^2 - 22x + 1 = 15 is x = 11 ± 3√15.