Final answer:
To solve the quadratic equation x^2 - 2x - 11 = 0 by completing the square, we add 1 to both sides to make the left side a perfect square. After taking the square root of both sides, we find the solutions x = 1 + √12 and x = 1 - √12, which are approximately 4.5 and -2.5.
Step-by-step explanation:
To solve the equation x^2 - 2x - 11 = 0 by completing the square, we first ensure the equation is in the form ax^2 + bx + c = 0. Here, a=1, b=-2, and c=-11. The next step is to move the constant term to the other side of the equation:
x^2 - 2x = 11
Now we find the number that completes the square for the x-terms. This number is (b/2)^2, so we calculate (-2/2)^2 = 1. We add this to both sides of the equation:
x^2 - 2x + 1 = 11 + 1
(x - 1)^2 = 12
Now we take the square root of both sides:
x - 1 = ±√12
Finally, we solve for x:
x = 1 ± √12
Which gives us the two solutions:
x = 1 + √12 or x = 1 - √12
After calculation, the solutions to one decimal place are:
x = 4.5 or x = -2.5
These values are the roots of the equation when solved by completing the square.