Complete the square to solve the following quadratic equation: x2−22x+1=15 .(1 point) Responses x=11+14−−√, 11−14−−√ x equals 11 plus Start Root 14 End Root comma 11 minus Start Root 14 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+315−−√, −11−315−−√
To complete the square, we need to rearrange the equation to isolate the quadratic term and then take half of the coefficient of the linear term squared.
Starting with the equation x^2 - 22x + 1 = 15, subtracting 1 from both sides gives us x^2 - 22x = 14.
To complete the square, we take half of the coefficient of the linear term, which is -22, and square it to get (-22/2)^2 = 11^2 = 121.
To balance the equation, we add 121 to both sides, resulting in x^2 - 22x + 121 = 14 + 121, or (x-11)^2 = 135.
Now we take the square root of both sides to solve for x, giving us x-11 = ±√135.
Adding 11 to both sides yields x = 11 ± √135.
So, the correct response is x = 11 ± √135.
To complete the square and solve the equation x^2 - 22x + 1 = 15, follow these steps:
Step 1: Move the constant term to the right side:
x^2 - 22x = 15 - 1
x^2 - 22x = 14
Step 2: Take half of the coefficient of x, square it, and add it to both sides:
(x - 11)^2 = 14 + 11^2
(x - 11)^2 = 14 + 121
(x - 11)^2 = 135
Step 3: Take the square root of both sides (remember to consider both positive and negative roots):
x - 11 = ±√135
Step 4: Solve for x:
x = 11 ± √135
So, the solutions to the quadratic equation x^2 - 22x + 1 = 15 after completing the square are:
x = 11 + √135
x = 11 - √135
To complete the square and solve the given quadratic equation, follow these steps:
Step 1: Move the constant term (15) to the right-hand side of the equation:
x^2 - 22x + 1 = 15 becomes x^2 - 22x + 1 - 15 = 0, which simplifies to:
x^2 - 22x - 14 = 0.
Step 2: Take the coefficient of x (which is -22) and divide it by 2, then square the result:
(-22 / 2)^2 = (-11)^2 = 121.
This value (121) will be used to complete the square.
Step 3: Add the value obtained in Step 2 (121) to both sides of the equation:
x^2 - 22x - 14 + 121 = 0 + 121, which simplifies to:
x^2 - 22x + 107 = 0.
Step 4: Rewrite the left-hand side of the equation as a perfect square trinomial:
(x - 11)^2 = 107.
Step 5: Take the square root of both sides of the equation while considering both positive and negative square roots:
x - 11 = ±√107.
Step 6: Solve for x by adding 11 to both sides of the equation:
x = 11 ±√107.
So, the solution to the given quadratic equation, after completing the square, is:
x = 11 + √107 and x = 11 - √107.