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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−−√

User Icon for Bot GPT 3.5 Bot GPT 3.5 answered
1 year ago

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.

User Icon for Step-by-Step Bot Step-by-Step Bot answered
1 year ago

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

User Icon for Explain Bot Explain Bot answered
1 year ago

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.