THe value of x which satisfy the equation `(sqrt(5x^2-8x+3))-sqrt((5x^2-9x+4))=sqrt((2x^2-2x))-sqrt((2x^2-3x+1))` is

Options

A. 3
B. 2
C. 1
D. 0

Explanation

To solve the equation

\[ \sqrt{5x^2 - 8x + 3} - \sqrt{5x^2 - 9x + 4} = \sqrt{2x^2 - 2x} - \sqrt{2x^2 - 3x + 1} \]

we will follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the equation to set it to zero:

\[ \sqrt{5x^2 - 8x + 3} - \sqrt{5x^2 - 9x + 4} - \sqrt{2x^2 - 2x} + \sqrt{2x^2 - 3x + 1} = 0 \]

### Step 2: Simplifying Each Square Root
Next, we simplify each square root expression:

1. \(5x^2 - 8x + 3\)
2. \(5x^2 - 9x + 4\)
3. \(2x^2 - 2x\)
4. \(2x^2 - 3x + 1\)

### Step 3: Factoring the Expressions
We can factor the expressions under the square roots:

1. \(5x^2 - 8x + 3 = (5x - 3)(x - 1)\)
2. \(5x^2 - 9x + 4 = (5x - 4)(x - 1)\)
3. \(2x^2 - 2x = 2x(x - 1)\)
4. \(2x^2 - 3x + 1 = (2x - 1)(x - 1)\)

### Step 4: Substitute Back into the Equation
Now substituting these factored forms back into the equation gives us:

\[ \sqrt{(5x - 3)(x - 1)} - \sqrt{(5x - 4)(x - 1)} - \sqrt{2x(x - 1)} + \sqrt{(2x - 1)(x - 1)} = 0 \]

### Step 5: Factor Out Common Terms
We can factor out \(\sqrt{x - 1}\):

\[ \sqrt{x - 1} \left( \sqrt{5x - 3} - \sqrt{5x - 4} - \sqrt{2x} + \sqrt{2x - 1} \right) = 0 \]

### Step 6: Setting Each Factor to Zero
From the product being zero, we have two cases:

1. \(\sqrt{x - 1} = 0\) which gives \(x - 1 = 0 \Rightarrow x = 1\)
2. The other factor \(\sqrt{5x - 3} - \sqrt{5x - 4} - \sqrt{2x} + \sqrt{2x - 1} = 0\) can be solved separately, but we will check if \(x = 1\) satisfies the original equation.

### Step 7: Check if \(x = 1\) Satisfies the Original Equation
Substituting \(x = 1\) into the original equation:

\[ \sqrt{5(1)^2 - 8(1) + 3} - \sqrt{5(1)^2 - 9(1) + 4} = \sqrt{2(1)^2 - 2(1)} - \sqrt{2(1)^2 - 3(1) + 1} \]

Calculating each term:

1. Left side: \(\sqrt{5 - 8 + 3} - \sqrt{5 - 9 + 4} = \sqrt{0} - \sqrt{0} = 0\)
2. Right side: \(\sqrt{2 - 2} - \sqrt{2 - 3 + 1} = \sqrt{0} - \sqrt{0} = 0\)

Both sides equal zero, confirming that \(x = 1\) is indeed a solution.

### Final Answer
Thus, the value of \(x\) that satisfies the equation is:

\[ \boxed{1} \]