To solve quadratic equations, you can use several methods. The most common ones are:

1. Factoring
2. Completing the Square
3. Using the Quadratic Formula
4. Graphing

1. Factoring

This method works when the quadratic equation can be factored into the product of two binomials. For example, solving $x^2 - 5x + 6 = 0$:

$x^2 - 5x + 6 = (x - 2)(x - 3) = 0$

Set each factor to zero and solve for $x$:

$x - 2 = 0 \quad \Rightarrow \quad x = 2$ $x - 3 = 0 \quad \Rightarrow \quad x = 3$

So, the solutions are $x = 2$ and $x = 3$.

2. Completing the Square

This method involves rewriting the quadratic equation in the form $(x - p)^2 = q$. For example, solving $x^2 - 4x - 5 = 0$:

$x^2 - 4x = 5$

Add 4 to both sides to complete the square:

$x^2 - 4x + 4 = 9$

Rewrite the left side as a square:

$(x - 2)^2 = 9$

Take the square root of both sides:

$x - 2 = \pm 3$

Solve for $x$:

$x = 2 \pm 3$

So, the solutions are $x = 5$ and $x = -1$.

3. Using the Quadratic Formula

The quadratic formula can solve any quadratic equation of the form $ax^2 + bx + c = 0$:

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

For example, solving $2x^2 - 4x - 6 = 0$:

$a = 2, \quad b = -4, \quad c = -6$

Plug these values into the formula:

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot (-6)}}{2 \cdot 2}$

Simplify inside the square root and solve:

$x = \frac{4 \pm \sqrt{16 + 48}}{4} = \frac{4 \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4}$

So, the solutions are:

$x = \frac{4 + 8}{4} = 3 \quad \text{and} \quad x = \frac{4 - 8}{4} = -1$

4. Graphing

You can graph the quadratic equation and find the points where it intersects the x-axis. These points are the solutions.

Would you like any more details or have any other questions? Here are some related questions you might find interesting:

1. Can you explain how to factor quadratic equations in more detail?
2. How do you graph quadratic functions?
3. What are the applications of the quadratic formula?
4. How do you use the discriminant to determine the nature of the roots?
5. Can you provide more examples of completing the square?

Tip: Always check if a quadratic equation can be factored first; it can often save time compared to other methods.