Find All Complex Solutions x^2=19x+1

$x^{2} = 19 x + 1$

Step 1

Move all the expressions to the left side of the equation.

Subtract $19 x$ from both sides of the equation.

$x^{2} - 19 x = 1$

Subtract $1$ from both sides of the equation.

$x^{2} - 19 x - 1 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 1$ , $b = - 19$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{19 \pm \sqrt{{(- 19)}^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 1$ .

Multiply $- 4$ by $1$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot - 1}}{2 \cdot 1}$

Multiply $- 4$ by $- 1$ .

$x = \frac{19 \pm \sqrt{361 + 4}}{2 \cdot 1}$

Add $361$ and $4$ .

$x = \frac{19 \pm \sqrt{365}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{19 \pm \sqrt{365}}{2}$

Step 5

The final answer is the combination of both solutions.

$x = \frac{19 + \sqrt{365}}{2}, \frac{19 - \sqrt{365}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{19 + \sqrt{365}}{2}, \frac{19 - \sqrt{365}}{2}$

Decimal Form:

$x = 19.05248658 \dots, - 0.05248658 \dots$