The equation `(2x^(2))/(x-1)-(2x +7)/(3) +(4-6x)/(x-1) +1=0` has the roots-

Options

A. 4 and 1
B. only 1
C. only 4
D. Neither 4 nor 1

Answer

To solve the equation

\[ \frac{2x^2}{x-1} - \frac{2x + 7}{3} + \frac{4 - 6x}{x-1} + 1 = 0, \]

we will follow these steps:

### Step 1: Identify Restrictions
First, we note that \( x \neq 1 \) because it would make the denominator zero, leading to an undefined expression.

### Step 2: Find a Common Denominator
The common denominator for the fractions is \( 3(x - 1) \). We will rewrite each term with this common denominator:

1. For \( \frac{2x^2}{x-1} \):
\[ \frac{2x^2}{x-1} = \frac{2x^2 \cdot 3}{3(x-1)} = \frac{6x^2}{3(x-1)} \]

2. For \( -\frac{2x + 7}{3} \):
\[ -\frac{2x + 7}{3} = -\frac{(2x + 7)(x - 1)}{3(x - 1)} = -\frac{(2x^2 + 7 - 2x - 7)}{3(x - 1)} = -\frac{2x^2 + 7 - 2x + 7}{3(x - 1)} \]

3. For \( \frac{4 - 6x}{x-1} \):
\[ \frac{4 - 6x}{x-1} = \frac{(4 - 6x) \cdot 3}{3(x-1)} = \frac{12 - 18x}{3(x-1)} \]

4. For \( 1 \):
\[ 1 = \frac{3(x-1)}{3(x-1)} = \frac{3x - 3}{3(x-1)} \]

### Step 3: Combine the Terms
Now we combine all the terms over the common denominator:

\[ \frac{6x^2 - (2x^2 - 2x + 7) + (12 - 18x) + (3x - 3)}{3(x-1)} = 0 \]

This simplifies to:

\[ 6x^2 - 2x^2 + 2x - 7 + 12 - 18x + 3x - 3 = 0 \]

Combining like terms:

\[ (6x^2 - 2x^2) + (2x - 18x + 3x) + (-7 + 12 - 3) = 0 \]

This results in:

\[ 4x^2 - 13x + 2 = 0 \]

### Step 4: Solve the Quadratic Equation
Now we can solve the quadratic equation \( 4x^2 - 13x + 2 = 0 \) using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Where \( a = 4, b = -13, c = 2 \):

\[ b^2 - 4ac = (-13)^2 - 4 \cdot 4 \cdot 2 = 169 - 32 = 137 \]

Now substituting into the formula:

\[ x = \frac{13 \pm \sqrt{137}}{8} \]

### Step 5: Identify the Roots
The roots of the equation are:

\[ x_1 = \frac{13 + \sqrt{137}}{8}, \quad x_2 = \frac{13 - \sqrt{137}}{8} \]

### Conclusion
Since \( x \neq 1 \), we check if either root equals 1. However, neither root will equal 1, hence both roots are valid.

### Final Answer
The roots of the equation are:

\[ x = \frac{13 + \sqrt{137}}{8} \quad \text{and} \quad x = \frac{13 - \sqrt{137}}{8} \]

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