Quadratic equations are equations of the form:

$ax^2 + bx + c = 0$

where $a$, $b$, and $c$ are constants, and $x$ represents the variable we want to solve for. The equation is "quadratic" because the highest exponent of $x$ is 2.

Key Methods for Solving Quadratic Equations:

1. Factoring: This method involves expressing the quadratic equation as a product of two binomials. Example: $x^2 + 5x + 6 = 0 \quad \Rightarrow \quad (x + 2)(x + 3) = 0$ The solutions would be $x = -2$ and $x = -3$.

2. Quadratic Formula: The quadratic formula can be used to find the solutions of any quadratic equation: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ This formula is derived from completing the square on the general form of a quadratic equation.

3. Completing the Square: This involves rewriting the quadratic equation in the form $(x + p)^2 = q$, and then solving for $x$. Example: $x^2 + 6x + 5 = 0 \quad \Rightarrow \quad (x + 3)^2 = 4$ Then solve $x + 3 = \pm 2$, giving $x = -1$ or $x = -5$.

4. Graphing: The quadratic equation can be represented as a parabola on a graph. The solutions correspond to the points where the parabola intersects the x-axis (the roots).

5. Square Root Method: If the quadratic is of the form $ax^2 = d$, then taking the square root of both sides can provide the solution. Example: $x^2 = 16 \quad \Rightarrow \quad x = \pm 4$

Would you like details on any specific method or an example?

Here are 5 related questions:

1. How do you apply the quadratic formula?
2. Can all quadratic equations be factored?
3. What is the discriminant in a quadratic equation?
4. How do you complete the square for any quadratic equation?
5. How do you graph a quadratic equation?

Tip: The discriminant, $b^2 - 4ac$, tells you the number and type of solutions—real or complex.