Question

solve $6x^2 - 2x + 1 = 0$ using the quadratic formula.

$x = \square$ and $x = \square$

Explanation:

Step1: Recall Quadratic Formula

For a quadratic equation \(ax^2 + bx + c = 0\), the solutions are given by \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 6\), \(b=-2\), \(c = 1\).

Step2: Calculate Discriminant (\(D=b^2 - 4ac\))

Substitute \(a = 6\), \(b=-2\), \(c = 1\) into \(D\):
\(D=(-2)^2-4\times6\times1=4 - 24=-20\).

Step3: Substitute into Quadratic Formula

\(x=\frac{-(-2)\pm\sqrt{-20}}{2\times6}=\frac{2\pm\sqrt{20}i}{12}\) (since \(\sqrt{-20}=\sqrt{20}i = 2\sqrt{5}i\)).
Simplify: \(x=\frac{2\pm2\sqrt{5}i}{12}=\frac{1\pm\sqrt{5}i}{6}\).

Answer:

\(x=\frac{1 + \sqrt{5}i}{6}\) and \(x=\frac{1 - \sqrt{5}i}{6}\)