Full solution

4

.) The solution to

x^{2}-4 x-6=0

in simplest surd form is

Identify Equation Type: Identify the type of equation. $\newline$ We have a quadratic equation in the form $ax^2 + bx + c = 0$ , where $a = 1$ , $b = -4$ , and $c = -6$ .
Apply Quadratic Formula: Apply the quadratic formula to find the roots of the equation. $\newline$ The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . $\newline$ Here, $a = 1$ , $b = -4$ , and $c = -6$ .
Calculate Discriminant: Calculate the discriminant $b^2 - 4ac$ . $\newline$ Discriminant = $(-4)^2 - 4(1)(-6) = 16 + 24 = 40$ .
Substitute Values: Substitute the values into the quadratic formula. $\newline$ $x = \frac{-(-4) \pm \sqrt{40}}{2 \cdot 1}$ $\newline$ $x = \frac{4 \pm \sqrt{40}}{2}$
Simplify Square Root: Simplify the square root of the discriminant. $\sqrt{40}$ can be written as $\sqrt{4\times10}$ , which simplifies to $2\times\sqrt{10}$ .
Substitute Simplified Root: Substitute the simplified square root back into the formula. $x = \frac{4 \pm 2\sqrt{10}}{2}$
Simplify Expression: Simplify the expression by dividing all terms by $\frac{2}{2}$ . $x = \frac{2 \pm \sqrt{10}}{2}$