Question

use the quadratic formula to solve for x.
$9x^2 + 3x - 1 = 0$
(if there is more than one solution, separate them with commas.)
$x = \square$

Explanation:

Step1: Identify coefficients

For quadratic equation \(ax^2 + bx + c = 0\), here \(a = 9\), \(b = 3\), \(c = -1\).

Step2: Recall quadratic formula

Quadratic formula is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\).

Step3: Calculate discriminant

Compute \(b^2 - 4ac=(3)^2 - 4\times9\times(-1)=9 + 36 = 45\).

Step4: Substitute into formula

Substitute \(a = 9\), \(b = 3\), \(b^2 - 4ac = 45\) into formula:
\(x=\frac{-3\pm\sqrt{45}}{2\times9}=\frac{-3\pm3\sqrt{5}}{18}=\frac{-1\pm\sqrt{5}}{6}\)

Answer:

\(\frac{-1 + \sqrt{5}}{6},\frac{-1 - \sqrt{5}}{6}\)