Solve the equation. $$ \frac{2 x}{x-2}=5+\frac{13 x^{2}}{x-2} $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\frac{2x}{x-2} = 5 + \frac{13x^2}{x-2}
$$

**Step 1: Combine the Fractions**

First, subtract $$5$$ from both sides to group the terms with the common denominator:

$$
\frac{2x}{x-2} - 5 = \frac{13x^2}{x-2}
$$

To combine the fractions, express $$5$$ with the common denominator $$x - 2$$:

$$
\frac{2x}{x-2} - \frac{5(x-2)}{x-2} = \frac{13x^2}{x-2}
$$

Now, combine the numerators:

$$
\frac{2x - 5(x - 2)}{x-2} = \frac{13x^2}{x-2}
$$

Simplify the numerator:

$$
2x - 5x + 10 = -3x + 10
$$

So the equation becomes:

$$
\frac{-3x + 10}{x-2} = \frac{13x^2}{x-2}
$$

**Step 2: Eliminate the Denominator**

Since $$x \neq 2$$ (to avoid division by zero), multiply both sides by $$x - 2$$:

$$
-3x + 10 = 13x^2
$$

**Step 3: Rearrange into Standard Quadratic Form**

Bring all terms to one side:

$$
13x^2 + 3x - 10 = 0
$$

**Step 4: Solve the Quadratic Equation**

Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$, where $$a = 13$$, $$b = 3$$, and $$c = -10$$:

$$
x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 13 \cdot (-10)}}{2 \cdot 13} = \frac{-3 \pm \sqrt{529}}{26} = \frac{-3 \pm 23}{26}
$$

This gives two solutions:

$$
x = \frac{20}{26} = \frac{10}{13} \quad \text{and} \quad x = \frac{-26}{26} = -1
$$

**Step 5: Verify the Solutions**

Ensure that neither solution makes the denominator $$x - 2$$ zero. Both $$x = \frac{10}{13}$$ and $$x = -1$$ satisfy this condition.

**Final Answer:**

The solutions are $$ x = \dfrac{10}{13} $$ and $$ x = -1 $$.
The solutions are $$ x = \frac{10}{13} $$ and $$ x = -1 $$.

Solve the equation. \[ \frac{2 x}{x-2}=5+\frac{13 x^{2}}{x-2} \]