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!