Question

Solve the equation. $$x^{2}-2 x-4=0$$

Short answer

The solutions for the equation are \(x = 1 + \sqrt{5}\) and \(x = 1 - \sqrt{5}\).

Step-by-step solution

Identify a, b, and c

Here the equation is \(x^{2}-2 x -4 = 0\). So, from the equation we identify \(a = 1\), the coefficient of \(x^2\), \(b = -2\), the coefficient of x, and \(c = -4\), the constant term.

Substitute into the Quadratic Formula

Substitute a, b, and c into the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Thus we have \(x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4*1*(-4)}}{2*1}\). Simplifying this we get \(x = \frac{2 \pm \sqrt{20}}{2}\)

Simplifying Further

Simplify the equation further. The square root of 20 simplifies to \(2\sqrt{5}\). Thus we have \(x = 1 \pm \sqrt{5}\)