The problem is to solve the equation $x(x-5)-3=19$ for $x$.

1. Problem Description

The problem is to solve the equation $x(x-5)-3=19$ for $x$.

2. Solution Steps

First, we simplify the equation:

$x(x-5) - 3 = 19$

$x^2 - 5x - 3 = 19$

Subtract 19 from both sides to set the equation to zero:

$x^2 - 5x - 3 - 19 = 0$

$x^2 - 5x - 22 = 0$

Now, we use the quadratic formula to solve for $x$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a=1$, $b=-5$, and $c=-22$.

$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-22)}}{2(1)}$

$x = \frac{5 \pm \sqrt{25 + 88}}{2}$

$x = \frac{5 \pm \sqrt{113}}{2}$

Therefore, the solutions are:

$x = \frac{5 + \sqrt{113}}{2}$ and $x = \frac{5 - \sqrt{113}}{2}$

3. Final Answer

$x = \frac{5 + \sqrt{113}}{2}$ or $x = \frac{5 - \sqrt{113}}{2}$