We are asked to solve the quadratic equation:

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

To solve for $x$, we can use the quadratic formula, which is given by:

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

In this case, the equation is in the form $ax^2 + bx + c = 0$, where:

• $a = 1$
• $b = -2$
• $c = -3$

Now, plug these values into the quadratic formula:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-3)}}{2(1)}$

Simplify the terms:

$x = \frac{2 \pm \sqrt{4 + 12}}{2}$ $x = \frac{2 \pm \sqrt{16}}{2}$ $x = \frac{2 \pm 4}{2}$

This gives two possible solutions:

1. $x = \frac{2 + 4}{2} = \frac{6}{2} = 3$
2. $x = \frac{2 - 4}{2} = \frac{-2}{2} = -1$

So the solutions are:

$x = 3 \quad \text{or} \quad x = -1$

Do you want a detailed explanation of any of the steps, or have further questions?

Here are 5 related questions you might find interesting:

1. What if the quadratic equation had no real solutions?
2. How can you factor the quadratic equation $x^2 - 2x - 3 = 0$ instead of using the quadratic formula?
3. What is the discriminant and how does it help in determining the number of solutions to a quadratic equation?
4. Can a quadratic equation have complex solutions? How?
5. What are some common methods for solving quadratic equations beyond the quadratic formula?

Tip: Always check the discriminant ($b^2 - 4ac$) to determine whether your quadratic equation has real or complex solutions.