To solve the equation 4 - x = (x - 2)^2, we can start by expanding the right side using the formula (a - b)^2 = a^2 - 2ab + b^2.
(x - 2)^2 = x^2 - 4x + 4
Now our equation becomes:
4 - x = x^2 - 4x + 4
Next, let's move all terms to one side to form a quadratic equation:
x^2 - 4x + 4 - 4 + x - 4 = 0
Simplifying, we get:
x^2 - 3x - 4 = 0
Now we have a quadratic equation in the form of ax^2 + bx + c = 0, where a = 1, b = -3, and c = -4.
To solve this quadratic equation, we can either factorize it or use the quadratic formula:
1. Factoring method:
(x - 4)(x + 1) = 0
x - 4 = 0 or x + 1 = 0
x = 4 or x = -1
2. Quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
x = (-(-3) ± √((-3)^2 - 4(1)(-4))) / (2(1))
x = (3 ± √(9 + 16)) / 2
x = (3 ± √25) / 2
x = (3 ± 5) / 2
This gives us two solutions: x = 4 and x = -1.
So the solution to the equation 2 + √(4 - x) = x is x = 4 and x = -1.