9х^2+4х-6=0

To solve the quadratic equation \(9x^2 + 4x - 6 = 0\), we can use the quadratic formula:

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

Where \(a = 9\), \(b = 4\), and \(c = -6\).

1. Calculate the discriminant (\(b^2 - 4ac\)):

\[ b^2 - 4ac = 4^2 - 4 \cdot 9 \cdot (-6) = 16 + 216 = 232 \]

2. Apply the values to the quadratic formula:

\[ x = \frac{-4 \pm \sqrt{232}}{2 \cdot 9} \]

3. Simplify \(\sqrt{232}\):

\[ \sqrt{232} = \sqrt{4 \cdot 58} = 2\sqrt{58} \]

4. Substitute back into the formula:

\[ x = \frac{-4 \pm 2\sqrt{58}}{18} \]

5. Simplify the fraction:

\[ x = \frac{-2 \pm \sqrt{58}}{9} \]

So the solutions to the equation \(9x^2 + 4x - 6 = 0\) are:

\[ x = \frac{-2 + \sqrt{58}}{9} \quad \text{and} \quad x = \frac{-2 - \sqrt{58}}{9} \]