Describe all of the solutions to the equation 2cos x + sqrt2=0?

Solve for "cos(x)"

the same way you would with any algebra problem.

2cos(x) = -√2

cos(x) = -(√2)/2

From then on, your answer depends on your understanding of "cos"

Because the value is negative, you know the solutions will all belong to quadrants 2 and 3.

2 cos x + sqrt 2 = 0

2 cos x = -sqrt 2

cos x = -sqrt 2 / 2

The solution for cos x = sqrt 2 / 2 is x = pi/4

cos x is negative in quadrants 2 and 3.

In quadrant 2, the angle whose reference angle is pi/4 is pi-pi/4 = 3pi/4 (one solution between 0 and 2pi)

In quadrant 3, the angle whose reference angle is pi/4 is pi+pi/4 = 5pi/4 (one solution between 0 and 2pi)

The period of cos x is 2pi

The general solution is:

x = 3pi/4 + 2n pi , where n is any integer

x = 5pi/4 + 2n pi , where n is any integer

cos x = - √2 / 2 = - 1 / √2

x = 135⁰ , 225⁰______quadrants 2 and 3

Solve it the same way I just showed you.