Question

Solve the quadratic equation. $$4 x^{2}-2 x-1=0$$

Short answer

The solutions to the quadratic equation are \(x = [1 ± sqrt(5)] / 4\)

Step-by-step solution

Identify values

Identify the values of a, b, and c in the equation. In this case, \(a = 4\), \(b = -2\), and \(c = -1\).

Apply the quadratic formula

Plug the values of a, b and c into the quadratic formula \(x = [-b ± sqrt(b ^2 - 4ac)] / 2a \). This leads to: \(x = [2 ± sqrt((-2) ^2 - 4*4*(-1))] / (2*4)\).

Simplify the equation

Simplify the equation further: \(x = [2 ± sqrt(4 + 16)] / 8\). This simplifies to \(x = [2 ± sqrt(20)] / 8\).

Calculate the final answer

Finally, simplify the above expression to get the roots of the quadratic. This results in two roots: \(x = [2 ± 2*sqrt(5)] / 8 = [1 ± sqrt(5)] / 4\)