The problem requires solving the quadratic equation $x^2 - 4x = 19$ by completing the square.

1. Problem Description

The problem requires solving the quadratic equation $x^2 - 4x = 19$ by completing the square.

2. Solution Steps

To complete the square for the equation $x^2 - 4x = 19$, we need to add a constant to both sides of the equation to make the left side a perfect square trinomial.

The coefficient of the $x$ term is $-4$. We take half of this coefficient, which is $\frac{-4}{2} = -2$, and square it: $(-2)^2 = 4$.

Now, we add 4 to both sides of the equation:

$x^2 - 4x + 4 = 19 + 4$

$x^2 - 4x + 4 = 23$

The left side is now a perfect square trinomial, which can be factored as $(x-2)^2$:

$(x-2)^2 = 23$

Now, we take the square root of both sides:

$\sqrt{(x-2)^2} = \pm\sqrt{23}$

$x - 2 = \pm\sqrt{23}$

Finally, we solve for $x$ by adding 2 to both sides:

$x = 2 \pm \sqrt{23}$

3. Final Answer

$x = 2 + \sqrt{23}, 2 - \sqrt{23}$