Let's solve the equation provided in the image:

$\frac{x - 7}{2x - 12} - 1 = \frac{-17}{2x}$

Step 1: Simplify the Equation

First, let's simplify the fraction on the left-hand side:

$\frac{x - 7}{2x - 12} = \frac{x - 7}{2(x - 6)}$

So the equation becomes:

$\frac{x - 7}{2(x - 6)} - 1 = \frac{-17}{2x}$

Step 2: Eliminate the Fraction on the Left-Hand Side

To eliminate the fraction on the left side, we'll subtract 1 from $\frac{x - 7}{2(x - 6)}$:

$\frac{x - 7 - 2(x - 6)}{2(x - 6)} = \frac{-17}{2x}$

Simplifying inside the numerator:

$\frac{x - 7 - 2x + 12}{2(x - 6)} = \frac{-17}{2x}$

$\frac{-x + 5}{2(x - 6)} = \frac{-17}{2x}$

Step 3: Cross Multiply

To solve for $x$, cross multiply the equation:

$(-x + 5) \cdot 2x = (-17) \cdot 2(x - 6)$

Expanding both sides:

$-2x^2 + 10x = -34x + 102$

Step 4: Move All Terms to One Side

Move all terms to the left side of the equation:

$-2x^2 + 44x - 102 = 0$

Step 5: Simplify and Solve the Quadratic Equation

Divide the entire equation by -2 to simplify:

$x^2 - 22x + 51 = 0$

Now, solve this quadratic equation using the quadratic formula:

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

where $a = 1$, $b = -22$, and $c = 51$.

$x = \frac{22 \pm \sqrt{(-22)^2 - 4(1)(51)}}{2(1)}$

$x = \frac{22 \pm \sqrt{484 - 204}}{2}$

$x = \frac{22 \pm \sqrt{280}}{2}$

$x = \frac{22 \pm \sqrt{4 \times 70}}{2}$

$x = \frac{22 \pm 2\sqrt{70}}{2}$

$x = 11 \pm \sqrt{70}$

Thus, the solutions are:

$x = 11 + \sqrt{70}, \quad x = 11 - \sqrt{70}$

Would you like more details or further clarification?

Here are five follow-up questions you might consider:

1. How do you apply the quadratic formula to equations with complex roots?
2. What is the significance of the discriminant in determining the nature of the roots?
3. How can you factor quadratic equations instead of using the quadratic formula?
4. What are other methods to solve quadratic equations?
5. How do the roots of a quadratic equation relate to its graph?

Tip: Always check if a quadratic equation can be simplified or factored before applying the quadratic formula. This can save time and make the solution easier.