Sure, let's talk about how to solve quadratic equations.

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \).

Here are the methods to solve a quadratic equation:

### 1. Factoring
If the quadratic can be factored, write it as \( (px + q)(rx + s) = 0 \), then solve for \( x \) by setting each factor to zero:
\[ px + q = 0 \quad \text{or} \quad rx + s = 0 \]

### 2. Using the Quadratic Formula
The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula always works, regardless of whether the quadratic is factorable.

### 3. Completing the Square
To complete the square, follow these steps:
1. Move the constant term to the other side of the equation: \( ax^2 + bx = -c \).
2. Divide by \( a \): \( x^2 + \frac{b}{a}x = -\frac{c}{a} \).
3. Add \(\left(\frac{b}{2a}\right)^2\) to both sides: \( x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = \left(\frac{b}{2a}\right)^2 - \frac{c}{a} \).
4. Write the left side as a square: \( \left( x + \frac{b}{2a} \right)^2 = \frac{b^2 - 4ac}{4a^2} \).
5. Solve for \( x \) by taking the square root of both sides.

### Example
Solve \( 2x^2 - 4x - 6 = 0 \) using the quadratic formula.

1. Identify \( a \), \( b \), and \( c \):
   \[ a = 2, \; b = -4, \; c = -6 \]

2. Plug into the quadratic formula:
   \[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot (-6)}}{2 \cdot 2} \]
   \[ x = \frac{4 \pm \sqrt{16 + 48}}{4} \]
   \[ x = \frac{4 \pm \sqrt{64}}{4} \]
   \[ x = \frac{4 \pm 8}{4} \]

3. Solve for \( x \):
   \[ x = \frac{4 + 8}{4} = 3 \quad \text{or} \quad x = \frac{4 - 8}{4} = -1 \]

So, the solutions are \( x = 3 \) and \( x = -1 \).

Would you like more details or have any questions?

Here are five more example questions you could ask:
1. How do you graph a quadratic function?
2. What are the applications of quadratic equations in real life?
3. How do you find the vertex of a quadratic function?
4. Can you explain the discriminant of a quadratic equation?
5. How do you solve quadratic equations by factoring?

**Tip:** Memorize the quadratic formula and understand each step to apply it effectively.

Explore how to solve quadratic equations using factoring, the quadratic formula, and completing the square. Includes detailed examples and step-by-step solutions. Suitable for high school students.

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