Find the roots of the equations.
Q. `(2x)/(x-4)+(2x-5)/(x-3)=8(1)/(2)`.

Answer

To solve the equation

\[ \frac{2x}{x-4} + \frac{2x-5}{x-3} = 8 \frac{1}{2} \]

we will follow these steps:

### Step 1: Convert Mixed Number to Improper Fraction
Convert \(8 \frac{1}{2}\) to an improper fraction:

\[ 8 \frac{1}{2} = \frac{17}{2} \]

### Step 2: Find the LCM of the Denominators
The denominators on the left-hand side are \(x-4\) and \(x-3\). The least common multiple (LCM) is:

\[ \text{LCM} = (x-4)(x-3) \]

### Step 3: Rewrite the Equation
Multiply both sides by the LCM to eliminate the fractions:

\[ \left(2x \cdot (x-3) + (2x-5) \cdot (x-4)\right) = \frac{17}{2} \cdot (x-4)(x-3) \]

### Step 4: Expand Both Sides
Expand the left-hand side:

\[ 2x(x-3) + (2x-5)(x-4) \]

Calculating each term:

\[ = 2x^2 - 6x + (2x^2 - 8x - 5x + 20) \]
\[ = 2x^2 - 6x + 2x^2 - 13x + 20 \]
\[ = 4x^2 - 19x + 20 \]

Now expand the right-hand side:

\[ \frac{17}{2}(x^2 - 7x + 12) \]
\[ = \frac{17}{2}(x^2 - 7x + 12) = \frac{17x^2 - 119x + 204}{2} \]

### Step 5: Clear the Fraction
Multiply both sides by 2 to eliminate the fraction:

\[ 2(4x^2 - 19x + 20) = 17x^2 - 119x + 204 \]
\[ 8x^2 - 38x + 40 = 17x^2 - 119x + 204 \]

### Step 6: Rearrange the Equation
Bring all terms to one side:

\[ 8x^2 - 38x + 40 - 17x^2 + 119x - 204 = 0 \]
\[ -9x^2 + 81x - 164 = 0 \]

Multiply through by -1 to simplify:

\[ 9x^2 - 81x + 164 = 0 \]

### Step 7: Use the Quadratic Formula
The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{D}}{2a} \]

where \(D = b^2 - 4ac\). Here, \(a = 9\), \(b = -81\), and \(c = 164\).

### Step 8: Calculate the Discriminant
Calculate \(D\):

\[ D = (-81)^2 - 4 \cdot 9 \cdot 164 \]
\[ = 6561 - 5904 = 657 \]

### Step 9: Substitute into the Quadratic Formula
Now substitute into the formula:

\[ x = \frac{81 \pm \sqrt{657}}{18} \]

### Step 10: Simplify the Roots
The square root of 657 can be simplified:

\[ \sqrt{657} = 3\sqrt{73} \]

Thus, the roots are:

\[ x = \frac{81 \pm 3\sqrt{73}}{18} \]
\[ = \frac{27 \pm \sqrt{73}}{6} \]

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

\[ x_1 = \frac{27 + \sqrt{73}}{6}, \quad x_2 = \frac{27 - \sqrt{73}}{6} \]