Question

solve $6x^2 - 2x + 1 = 0$ using any method.

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

Explanation:

Step1: Identify coefficients

For quadratic equation \(ax^2 + bx + c = 0\), here \(a = 6\), \(b = -2\), \(c = 1\).

Step2: Calculate discriminant

Discriminant \(D = b^2 - 4ac\). Substitute values: \(D = (-2)^2 - 4\times6\times1 = 4 - 24 = -20\).

Step3: Use quadratic formula

Quadratic formula: \(x = \frac{-b \pm \sqrt{D}}{2a}\). Since \(D = -20 = 20i^2\) (where \(i = \sqrt{-1}\)), \(\sqrt{D} = \sqrt{20}i = 2\sqrt{5}i\).
Substitute \(a\), \(b\), \(\sqrt{D}\) into formula: \(x = \frac{-(-2) \pm 2\sqrt{5}i}{2\times6} = \frac{2 \pm 2\sqrt{5}i}{12} = \frac{1 \pm \sqrt{5}i}{6}\).

Answer:

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