How do I solve this equation: square root of x minus 4 = 2 minus x

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I see two possible problems here.

➊ As written:
   √ ̅x̅ ̅  - 4 = 2 - x
        √ ̅x̅ ̅ = 6 - x            ← Now, square both sides
    (√ ̅x̅ ̅ )² = (6-x)²            ← Because we squared both sides, we must
                                          check for extraneous roots
           x = x² - 12x + 36
           0 = x² - 13x + 36
           0 = (x-4)(x-9)
       possible solutions are
           x = 4 and x = 9
Checking for extraneous roots shows us that x=9 doesn't work.

               ANSWER
                  x = 4

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➋ Figuring for omitted parentheses:
    √ ̅x̅-̅4̅ ̅  = 2 - x     ← Observation: Square roots are real only if the radicand
                              is positive or zero. So, it must be that x-4≥0 which requires x≥4.
                              Also, square roots are never negative. So 2-x must be positive
                              or zero. So, it must be also that 2-x≥0 which requires x≤2.

                              Therefore, solutions for x must be both ≥4 and ≤2.
                              Since a real number cannot satisfy both conditions, there is
                              no real solution to √ ̅x̅-̅4̅ ̅  = 2-x

                                                            ANSWER
                                                        no real solution

Have a good one!
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first you square each side of the equation

x - 4 = 4 - x^2

then put all numbers and variables on one side

x^2 + x - 8

it cannot factor so you use the quadratic formula

x = -1 + or - sq.rt.1 + 32 / 2

x = -1 + or - sq.rt.33 / 2

i am sure you can figure it out from there

Square both sides.

x - 4 = (2 - x)^2