9x^2-7x-2=0

To solve the quadratic equation \(9x^2 - 7x - 2 = 0\), we can use the quadratic formula:

The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In this equation, \(a = 9\), \(b = -7\), and \(c = -2\).

Substitute these values into the formula:
\[ x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(9)(-2)}}{2(9)} \]

Simplify:
\[ x = \frac{7 \pm \sqrt{49 + 72}}{18} \]
\[ x = \frac{7 \pm \sqrt{121}}{18} \]
\[ x = \frac{7 \pm 11}{18} \]

Now we have two possible solutions:
1. When using the plus sign:
\[ x = \frac{7 + 11}{18} = \frac{18}{18} = 1 \]

2. When using the minus sign:
\[ x = \frac{7 - 11}{18} = \frac{-4}{18} = -\frac{2}{9} \]

Therefore, the solutions to the equation \(9x^2 - 7x - 2 = 0\) are \(x = 1\) and \(x = -\frac{2}{9}\).