Final answer:
To solve the equation x^2 + 6x = 13 by completing the square and factoring, add 9 to both sides to form a perfect square, which results in x = -3 ±√22 after taking the square root and isolating x.
Step-by-step explanation:
The equation x^2 + 6x = 13 can be solved by completing the square and then factoring. First, the equation is rewritten by adding a number to both sides to make the left side a perfect square.
To complete the square, take half of the coefficient of x, which is 6, and square it, giving us (6/2)^2 = 9. Add 9 to both sides of the equation to get:
x^2 + 6x + 9 = 13 + 9
Now, the equation becomes:
(x + 3)^2 = 22
The left side of the equation is a perfect square. To solve for x, take the square root of both sides:
x + 3 = ±√22
Finally, subtract 3 from both sides to isolate x:
x = -3 ±√22
Thus, we have two real solutions for x, based on the positive and negative square roots of 22.