To solve this equation, we need to isolate the square roots on one side and square both sides to eliminate the square roots. Let's begin with the given equation:
√(4x - 2) + √(3x - 3) = √(x + 1)
Step 1: Move √(3x - 3) to the right side.
√(4x - 2) = √(x + 1) - √(3x - 3)
Step 2: Square both sides.
(√(4x - 2))^2 = (√(x + 1) - √(3x - 3))^2
4x - 2 = (x + 1) - 2√((x + 1)(3x - 3)) + (3x - 3)
Step 3: Simplify the equation.
4x - 2 = x + 1 - 2√(3x^2 - 2x - 3) + 3x - 34x - 2 = 4x - 2 - 2√(3x^2 - 2x - 3)
Step 4: Combine like terms.
0 = -2√(3x^2 - 2x - 3)
Step 5: Square both sides again.
0 = 4(3x^2 - 2x - 3)0 = 12x^2 - 8x - 12
Step 6: Divide by 4.
0 = 3x^2 - 2x - 3
Step 7: Factor the quadratic equation.
0 = (3x + 1)(x - 3)
Step 8: Solve for x.
3x + 1 = 03x = -1x = -1/3
x - 3 = 0x = 3
Therefore, the solutions to the given equation are x = -1/3 and x = 3.
To solve this equation, we need to isolate the square roots on one side and square both sides to eliminate the square roots. Let's begin with the given equation:
√(4x - 2) + √(3x - 3) = √(x + 1)
Step 1: Move √(3x - 3) to the right side.
√(4x - 2) = √(x + 1) - √(3x - 3)
Step 2: Square both sides.
(√(4x - 2))^2 = (√(x + 1) - √(3x - 3))^2
4x - 2 = (x + 1) - 2√((x + 1)(3x - 3)) + (3x - 3)
Step 3: Simplify the equation.
4x - 2 = x + 1 - 2√(3x^2 - 2x - 3) + 3x - 3
4x - 2 = 4x - 2 - 2√(3x^2 - 2x - 3)
Step 4: Combine like terms.
0 = -2√(3x^2 - 2x - 3)
Step 5: Square both sides again.
0 = 4(3x^2 - 2x - 3)
0 = 12x^2 - 8x - 12
Step 6: Divide by 4.
0 = 3x^2 - 2x - 3
Step 7: Factor the quadratic equation.
0 = (3x + 1)(x - 3)
Step 8: Solve for x.
3x + 1 = 0
3x = -1
x = -1/3
x - 3 = 0
x = 3
Therefore, the solutions to the given equation are x = -1/3 and x = 3.