Solve $\color{blue}{x^2-4x-1 = 0}$ using the Quadratic Formula.
Step 1: Read the values of $ a $, $ b $, and $ c $ from the quadratic equation: $ a $ is the number in front of $ x^2 $, $ b $ is the number in front of $ x $, $ c $ is the number at the end. In our case:
$$ a = 1, \,\, b = -4, \,\, c = -1 $$Step 2: Plug in the values for $ a $, $ b $, and $ c $ into the quadratic formula.
$$ \begin{aligned} x_1,x_2 &= \frac{-b \pm \sqrt{b^2-4ac}}{2a} \\[1 em] x_1,x_2 &= \frac{ -(-4) \pm \sqrt{ (-4)^2 - 4 \cdot 1 \cdot (-1)} }{ 2 \cdot 1 } \end{aligned} $$Step 3: Simplify expression under the square root.
$$ x_1,x_2 = \frac{ 4 \pm \sqrt{ 20 } }{ 2 } $$Step 4: Solve for $ x $
$$ \begin{aligned} & \color{blue}{ x_1 = \frac{ 4~-~\sqrt{ 20 } }{ 2 } = 2-\sqrt{ 5 } } \\\\ & \color{blue}{ x_2 = \frac{ 4~+~\sqrt{ 20 } }{ 2 } = 2+\sqrt{ 5 } } \end{aligned} $$