Find the roots of the function
f(x)= x^2 +2x+2
a)-1 + sq root of 2 and -1 - sq root of 2
b) -1 + i and -1 - i
c)-1 -2i sq root of 2 and -1-2i sq root of 2
d)-1 +2i and -1-2i
The i's are in italics but I can't .make them in italic here for some reason
completing the square:
x^2+2x +1= -2 +1
(x+1)^2=-1
x+1= +-i sqrt2
solve for x
To find the roots of the given function f(x) = x^2 + 2x + 2, we can complete the square. Here's how you can do it:
Step 1: Start with the given function: f(x) = x^2 + 2x + 2.
Step 2: Group the first two terms together and leave the constant term separate: (x^2 + 2x) + 2.
Step 3: To complete the square, take half of the coefficient of the x-term, which is 2, and square it. (2/2)^2 = 1.
Step 4: Add and subtract the value obtained in Step 3 inside the parentheses: (x^2 + 2x + 1 - 1) + 2.
Step 5: Simplify and rearrange terms: (x + 1)^2 - 1 + 2.
Step 6: Combine like terms: (x + 1)^2 + 1.
Step 7: Set the expression equal to zero and solve for x: (x + 1)^2 + 1 = 0.
Step 8: Move the constant term to the other side: (x + 1)^2 = -1.
Step 9: Take the square root of both sides: √[(x + 1)^2] = ±√(-1).
Step 10: Simplify the square root: x + 1 = ±i√1.
Step 11: Further simplify: x + 1 = ±i.
Step 12: Solve for x: x = -1 + i and x = -1 - i.
Therefore, the correct answer is option b) -1 + i and -1 - i, which corresponds to the roots of the given function.