9х^2-5х=1

To solve the quadratic equation \( 9x^2 - 5x - 1 = 0 \), we can use the quadratic formula:

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

In this equation, \( a = 9 \), \( b = -5 \), and \( c = -1 \).

1. First, calculate the discriminant \( b^2 - 4ac \): \[ b^2 = (-5)^2 = 25 \] \[ 4ac = 4 \cdot 9 \cdot (-1) = -36 \] \[ b^2 - 4ac = 25 - (-36) = 25 + 36 = 61 \]

2. Now, substitute the values into the quadratic formula: \[ x = \frac{-(-5) \pm \sqrt{61}}{2 \cdot 9} \]

3. Simplify: \[ x = \frac{5 \pm \sqrt{61}}{18} \]

Thus, the solutions for the equation \( 9x^2 - 5x - 1 = 0 \) are: \[ x = \frac{5 + \sqrt{61}}{18} \quad \text{and} \quad x = \frac{5 - \sqrt{61}}{18} \]