Could you teach me quadratics and how you factorize them?

### Understanding Quadratics and Factorization

#### 1. **What is a Quadratic?**
A **quadratic equation** is a second-degree polynomial, typically written in the form:

\[
ax^2 + bx + c = 0
\]

Here:
- \(a\), \(b\), and \(c\) are constants (with \(a \neq 0\)).
- \(x\) represents the variable.

Examples:
- \(x^2 - 5x + 6 = 0\)
- \(2x^2 + 7x - 4 = 0\)

#### 2. **What Does Factorizing Mean?**
**Factorizing** a quadratic involves rewriting it as a product of two linear factors:
\[
ax^2 + bx + c = (px + q)(rx + s)
\]

Where:
- \(p\), \(q\), \(r\), and \(s\) are constants.
- Expanding the product gives the original quadratic.

#### 3. **Steps to Factorize Quadratics**

---

### Case 1: Simple Quadratics \((a = 1)\)
If \(a = 1\), the quadratic is:
\[
x^2 + bx + c
\]

#### Steps:
1. Find two numbers that:
   - Multiply to \(c\) (the constant term).
   - Add to \(b\) (the coefficient of \(x\)).

2. Write the factors as:
   \[
   x^2 + bx + c = (x + m)(x + n)
   \]

#### Example:
Factorize \(x^2 - 5x + 6\):
1. \(c = 6\), \(b = -5\).
2. Numbers are \(-2\) and \(-3\) because:
   - \((-2) \times (-3) = 6\)
   - \((-2) + (-3) = -5\).
3. Write as:
   \[
   x^2 - 5x + 6 = (x - 2)(x - 3)
   \]

---

### Case 2: Quadratics with \(a \neq 1\)
For quadratics like \(ax^2 + bx + c\), \(a > 1\):

#### Steps:
1. Multiply \(a \times c\).
2. Find two numbers that:
   - Multiply to \(a \cdot c\).
   - Add to \(b\).
3. Rewrite the middle term \(bx\) using these numbers.
4. Group terms and factorize in pairs.
5. Write the result as a product of two binomials.

#### Example:
Factorize \(2x^2 + 7x + 3\):
1. \(a \cdot c = 2 \times 3 = 6\), \(b = 7\).
2. Numbers are \(6\) and \(1\):
   - \(6 \cdot 1 = 6\), \(6 + 1 = 7\).
3. Rewrite \(7x\) as \(6x + x\):
   \[
   2x^2 + 6x + x + 3
   \]
4. Group terms:
   \[
   (2x^2 + 6x) + (x + 3)
   \]
5. Factorize each group:
   \[
   2x(x + 3) + 1(x + 3)
   \]
6. Factor out \((x + 3)\):
   \[
   (2x + 1)(x + 3)
   \]

---

### Case 3: Special Cases
- **Perfect Squares**: \(x^2 + 6x + 9 = (x + 3)^2\)
- **Difference of Squares**: \(x^2 - 16 = (x - 4)(x + 4)\)

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### Practice Problems:
Try these for yourself:
1. \(x^2 + 8x + 15\)
2. \(3x^2 + 11x + 10\)
3. \(x^2 - 9\)
4. \(2x^2 - 4x - 6\)
5. \(x^2 + 5x + 6\)

---

### Tip:
**Always check your factors by expanding them!** This ensures they simplify back to the original quadratic.

Learn the basics of quadratic equations and their factorization. This comprehensive guide covers algebraic concepts such as factoring simple quadratics, working with coefficients, and applying the zero product property. Suitable for students in Grades 8-10.

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