What is quadratic form and what are the formulas

### What is Quadratic Form?

A **quadratic form** is a polynomial of degree two in one or more variables. For a single variable \(x\), a quadratic expression is typically of the form:

\[
ax^2 + bx + c
\]

where:
- \(a\), \(b\), and \(c\) are constants,
- \(x\) is the variable,
- \(a \neq 0\) (because if \(a = 0\), it would no longer be a quadratic expression but a linear one).

### The Quadratic Equation

A **quadratic equation** is formed when a quadratic expression is set equal to zero:

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

### The Quadratic Formula

The quadratic formula is a method to find the roots (solutions) of the quadratic equation. It is derived by completing the square on the general quadratic equation:

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

Here:
- The expression \( \sqrt{b^2 - 4ac} \) is called the **discriminant**.
- The discriminant determines the nature of the roots:
  - If \( b^2 - 4ac > 0 \), there are two distinct real roots.
  - If \( b^2 - 4ac = 0 \), there is exactly one real root (a repeated root).
  - If \( b^2 - 4ac < 0 \), there are two complex (non-real) roots.

### Example of Using the Quadratic Formula

Given the quadratic equation:

\[
3x^2 - 5x + 2 = 0
\]

1. Identify \(a\), \(b\), and \(c\):
   \[
   a = 3, \quad b = -5, \quad c = 2
   \]

2. Plug the values into the quadratic formula:
   \[
   x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(2)}}{2(3)}
   \]
   \[
   x = \frac{5 \pm \sqrt{25 - 24}}{6}
   \]
   \[
   x = \frac{5 \pm \sqrt{1}}{6}
   \]

3. Simplify to find the roots:
   \[
   x = \frac{5 \pm 1}{6}
   \]
   So, the two roots are:
   \[
   x_1 = \frac{6}{6} = 1, \quad x_2 = \frac{4}{6} = \frac{2}{3}
   \]

### Other Formulas Related to Quadratic Equations

1. **Vertex Form**: A quadratic function can be written in vertex form as:
   \[
   y = a(x - h)^2 + k
   \]
   where \((h, k)\) is the vertex of the parabola.

2. **Factoring**: Sometimes, the quadratic can be factored directly:
   \[
   ax^2 + bx + c = (mx + n)(px + q)
   \]
   where \(m \times p = a\), \(n \times q = c\), and \(mp + nq = b\).

3. **Completing the Square**: A method to convert a quadratic equation into vertex form, which also helps derive the quadratic formula.

4. **Discriminant**: As mentioned earlier, it determines the nature of the roots:
   \[
   \text{Discriminant} = b^2 - 4ac
   \]

Would you like more examples or details on any of these concepts? Here are some related questions:

1. How do you derive the quadratic formula using completing the square?
2. What is the geometric interpretation of the vertex of a quadratic function?
3. How can you factor a quadratic equation when the discriminant is a perfect square?
4. What are some real-life applications of quadratic equations?
5. How does the sign of the leading coefficient \(a\) affect the shape of the parabola?

**Tip:** When solving quadratic equations, always check if factoring is possible before using the quadratic formula; it can save time!

Explore the concept of quadratic forms and equations with detailed formulas and examples. Learn how to solve quadratic equations using the quadratic formula and vertex form. Suitable for high school students.

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