3/x=2x+5

To solve the equation \( \frac{3}{x} = 2x + 5 \), follow these steps:

1. Eliminate the fraction by multiplying both sides by \( x \) (assuming \( x \neq 0 \)): \[ 3 = x(2x + 5) \]

2. Distribute on the right side: \[ 3 = 2x^2 + 5x \]

3. Rearrange the equation to set it to zero: \[ 2x^2 + 5x - 3 = 0 \]

4. Use the quadratic formula to solve for \( x \), where \( a = 2 \), \( b = 5 \), and \( c = -3 \): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

5. Calculate the discriminant: \[ b^2 - 4ac = 5^2 - 4 \cdot 2 \cdot (-3) = 25 + 24 = 49 \]

6. Substitute into the quadratic formula: \[ x = \frac{-5 \pm \sqrt{49}}{2 \cdot 2} \] \[ x = \frac{-5 \pm 7}{4} \]

7. Calculate the two potential solutions:

   • Using \( +7 \): \[ x = \frac{-5 + 7}{4} = \frac{2}{4} = \frac{1}{2} \]
   • Using \( -7 \): \[ x = \frac{-5 - 7}{4} = \frac{-12}{4} = -3 \]

8. Final solutions: The solutions to the equation \( \frac{3}{x} = 2x + 5 \) are: \[ x = \frac{1}{2} \quad \text{and} \quad x = -3 \]