To solve the quadratic equation \(3x^2 + 2x - 7 = 0\), we can use the quadratic formula:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]
where \(a = 3\), \(b = 2\), and \(c = -7\).
- Calculate the discriminant \(D = b^2 - 4ac\):
\[ D = 2^2 - 4 \cdot 3 \cdot (-7) = 4 + 84 = 88 \]
- Apply the quadratic formula:
\[ x = \frac{{-2 \pm \sqrt{88}}}{2 \cdot 3} \]
- Simplify \(\sqrt{88}\):
\[ \sqrt{88} = \sqrt{4 \cdot 22} = 2\sqrt{22} \]
- Substitute back into the formula:
\[ x = \frac{{-2 \pm 2\sqrt{22}}}{6} = \frac{{-1 \pm \sqrt{22}}}{3} \]
Thus, the solutions to the equation \(3x^2 + 2x - 7 = 0\) are:
\[ x = \frac{{-1 + \sqrt{22}}}{3} \quad \text{and} \quad x = \frac{{-1 - \sqrt{22}}}{3} \]