Giải phương trình

To solve the equation, we need to consider two cases: when 1-2x is positive and when it is negative.

Case 1: 1-2x ≥ 0
In this case, we can remove the absolute value sign and rewrite the equation as:
7 - 4√(2x-1) = 3(1-2x)

Simplifying the equation:
7 - 4√(2x-1) = 3 - 6x

Move the terms involving x to one side:
-4√(2x-1) + 6x = -4x + 3

Square both sides to eliminate the square root:
16(2x-1) + 36x^2 - 48x√(2x-1) = 16x^2 - 24x + 9

Expand and simplify:
32x - 16 + 36x^2 - 48x√(2x-1) = 16x^2 - 24x + 9

Rearrange the terms:
36x^2 - 16x^2 + 32x + 48x√(2x-1) - 24x + 48x√(2x-1) - 9 + 16 = 0

Combine like terms:
20x^2 + 56x√(2x-1) - 9 = 0

Case 2: 1-2x < 0
In this case, we need to consider the absolute value of 1-2x, so we rewrite the equation as:
7 - 4√(2x-1) = 3(2x-1)

Simplifying the equation:
7 - 4√(2x-1) = 6x - 3

Move the terms involving x to one side:
-4√(2x-1) - 6x = -10

Square both sides to eliminate the square root:
16(2x-1) + 36x^2 - 48x√(2x-1) = 100

Expand and simplify:
32x - 16 + 36x^2 - 48x√(2x-1) = 100

Rearrange the terms:
36x^2 - 16 + 32x + 48x√(2x-1) - 100 = 0

Combine like terms:
36x^2 + 32x + 48x√(2x-1) - 116 = 0

Now we have two quadratic equations to solve. We can solve them separately and find the values of x that satisfy each case.