To solve the quadratic equation $x^2 - 5x - 3 = 0$ by completing the square, follow these steps:

Step 1: Move the constant term to the other side

Start with the equation: $x^2 - 5x - 3 = 0$ Add 3 to both sides: $x^2 - 5x = 3$

Step 2: Complete the square

To complete the square, take half of the coefficient of $x$, square it, and add it to both sides of the equation. The coefficient of $x$ is $-5$, so half of $-5$ is $-\frac{5}{2}$, and squaring it gives: $\left(-\frac{5}{2}\right)^2 = \frac{25}{4}$ Add $\frac{25}{4}$ to both sides of the equation: $x^2 - 5x + \frac{25}{4} = 3 + \frac{25}{4}$

Step 3: Simplify both sides

The left side of the equation is now a perfect square trinomial: $\left(x - \frac{5}{2}\right)^2 = \frac{12}{4} + \frac{25}{4}$ Combine the fractions on the right side: $\left(x - \frac{5}{2}\right)^2 = \frac{37}{4}$

Step 4: Solve for $x$

Take the square root of both sides: $x - \frac{5}{2} = \pm \sqrt{\frac{37}{4}}$ Simplify the square root: $x - \frac{5}{2} = \pm \frac{\sqrt{37}}{2}$ Finally, solve for $x$ by adding $\frac{5}{2}$ to both sides: $x = \frac{5}{2} \pm \frac{\sqrt{37}}{2}$

Step 5: Express the final solutions

The solutions are: $x = \frac{5 + \sqrt{37}}{2} \quad \text{or} \quad x = \frac{5 - \sqrt{37}}{2}$

These are the same solutions obtained earlier using the quadratic formula.

Do you have any questions or need further clarification on this method?

Here are 5 related questions to explore further:

1. How does completing the square compare to using the quadratic formula for solving quadratic equations?
2. Can completing the square be used to derive the quadratic formula?
3. What is the geometric interpretation of completing the square on a graph?
4. How would the process change if the leading coefficient $a$ was not 1?
5. How can completing the square help in finding the vertex of a parabola?

Tip: Completing the square is a powerful method not only for solving quadratic equations but also for converting quadratic functions into vertex form, which is useful in graphing.