If `3x^(2)-4 sqrt((3x^(2)-4x+1))=4x-4,` then x= ….

Answer

To solve the equation \( 3x^2 - 4\sqrt{3x^2 - 4x + 1} = 4x - 4 \), we will follow these steps:

### Step 1: Isolate the square root
We start by moving the square root term to the right side of the equation:

\[ 3x^2 - 4 = 4x - 4\sqrt{3x^2 - 4x + 1} \]

Rearranging gives:

\[ 4\sqrt{3x^2 - 4x + 1} = 4x - 3x^2 + 4 \]

### Step 2: Simplify the equation
Dividing both sides by 4:

\[ \sqrt{3x^2 - 4x + 1} = x - \frac{3}{4} + 1 \]

This simplifies to:

\[ \sqrt{3x^2 - 4x + 1} = x - \frac{3}{4} \]

### Step 3: Square both sides
Next, we square both sides to eliminate the square root:

\[ 3x^2 - 4x + 1 = \left(x - \frac{3}{4}\right)^2 \]

Expanding the right side:

\[ 3x^2 - 4x + 1 = x^2 - \frac{3}{2}x + \frac{9}{16} \]

### Step 4: Rearranging the equation
Now, we will bring all terms to one side of the equation:

\[ 3x^2 - 4x + 1 - x^2 + \frac{3}{2}x - \frac{9}{16} = 0 \]

Combining like terms:

\[ (3x^2 - x^2) + (-4x + \frac{3}{2}x) + (1 - \frac{9}{16}) = 0 \]

This simplifies to:

\[ 2x^2 - \frac{5}{2}x + \frac{7}{16} = 0 \]

### Step 5: Clear the fraction
To eliminate the fraction, multiply through by 16:

\[ 32x^2 - 40x + 7 = 0 \]

### Step 6: Use the quadratic formula
Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 32, b = -40, c = 7 \):

Calculating the discriminant:

\[ b^2 - 4ac = (-40)^2 - 4 \cdot 32 \cdot 7 = 1600 - 896 = 704 \]

Now substituting into the quadratic formula:

\[ x = \frac{40 \pm \sqrt{704}}{64} \]

### Step 7: Simplify the square root
The square root of 704 can be simplified:

\[ \sqrt{704} = \sqrt{16 \cdot 44} = 4\sqrt{44} = 4\sqrt{4 \cdot 11} = 8\sqrt{11} \]

Thus, we have:

\[ x = \frac{40 \pm 8\sqrt{11}}{64} = \frac{5 \pm \sqrt{11}}{8} \]

### Step 8: Solve for the second case
Now, we also need to consider the case when \( y - 1 = 0 \):

Setting \( y = 1 \):

\[ 3x^2 - 4x + 1 = 1 \]

This simplifies to:

\[ 3x^2 - 4x = 0 \]

Factoring out \( x \):

\[ x(3x - 4) = 0 \]

Thus, \( x = 0 \) or \( x = \frac{4}{3} \).

### Final Solutions
The final solutions for \( x \) are:

\[ x = 0, \quad x = \frac{4}{3}, \quad x = \frac{5 + \sqrt{11}}{8}, \quad x = \frac{5 - \sqrt{11}}{8} \]