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: Use quadratic formula

Quadratic formula: \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\)
Calculate discriminant \(D = b^2 - 4ac = (-2)^2 - 4\times6\times1 = 4 - 24 = -20\)

Step3: Substitute into formula

\(x=\frac{-(-2)\pm\sqrt{-20}}{2\times6}=\frac{2\pm2i\sqrt{5}}{12}=\frac{1\pm i\sqrt{5}}{6}\)

Answer:

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