Question

Solve the quadratic equation. $$x^{2}-6 x-1=0$$

Short answer

The solutions of the quadratic equation are \(x=3+\sqrt{10}\) and \(x=3-\sqrt{10}\).

Step-by-step solution

Write down the given equation

The given equation is \(x^{2}-6x-1=0\). Identify the coefficients \(a=1\), \(b=-6\), and \(c=-1\).

Apply the Quadratic formula

The Quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Substitute coefficients into the formula to get \(x=\frac{-(-6)\pm\sqrt{(-6)^{2}-4(1)(-1)}}{2(1)}\). That simplifies to \(x=\frac{6\pm\sqrt{36+4}}{2}\).

Simplify under the root

Further simplify under the root \(\sqrt{36+4}\) to get \(\sqrt{40}\). The equation then becomes \(x=\frac{6\pm\sqrt{40}}{2}\).

Calculate the roots

Continue simplifying to get \(x=\frac{6+2\sqrt{10}}{2}\) or \(x=\frac{6-2\sqrt{10}}{2}\), which further simplifies to \(x=3+\sqrt{10}\) and \(x=3-\sqrt{10}\). Verify these by substituting back into the original equation and confirm that \(x^{2}-6x-1=0\).