Question

Solve the quadratic equation. $$x^{2}-4 x-6=0$$

Short answer

The solutions to the equation \( x^{2}-4x-6=0 \) are \( x=2+\sqrt{10} \) and \( x=2-\sqrt{10} \).

Step-by-step solution

Identify values of a, b and c

In the given equation \( x^{2}-4x-6=0 \), \( a=1 \), \( b=-4 \), and \( c=-6 \) are the coefficients of the quadratic equation.

Substitute values into the quadratic formula

Now, substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula \( x=\frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \). So \( x=\frac{-(-4) \pm \sqrt{(-4)^{2}-4*1*(-6)}}{2*1} \).

Simplifying the expression

Simplifying the expression gives the solution \( x=\frac{4 \pm \sqrt{16+24}}{2} \). Which further simplifies to \( x=\frac{4 \pm \sqrt{40}}{2} \). After simplification, we get \( x=\frac{4 \pm 2\sqrt{10}}{2} \). This simplifies further to \( x= 2 \pm \sqrt{10} \). Thus, the roots of the given quadratic equation are \( x=2+\sqrt{10} \) and \( x=2-\sqrt{10} \).