Arithmetic Course/Polynominal Equation

Polynomial Equation
An equation is an expression of one variable such that
 * $$f(x) = Ax^n + Bx^{(n-1)} + x^1 + x^0 = 0 .$$polynomials used to solve the theory of equations.

Solving Polynomial Equation
Solving polynomial equations involves finding all the values of variable x that satisfy f(x) = 0.

First Order Equation
A first order polynomial equation of one variable x has the general form
 * Ax + B = 0

Rewrite the equation above
 * $$x + \frac{B}{A} = 0$$
 * $$x = -\frac{B}{A}$$

Second Order Equation
A second order polynomial equation of one variable x has the general form
 * 1) $$Ax^2 + Bx + C = 0$$
 * 2) $$Ax^2 + C = 0$$
 * 3) $$Ax^2 - C = 0$$

Method 1
$$Ax^2 + Bx + C = 0$$
 * $$x^2 + \frac{B}{A}x + \frac{C}{A} = 0$$
 * $$x = -\alpha \pm \lambda$$

Where
 * $$\alpha = - \frac{B}{2A}$$
 * $$\beta = - \frac{C}{A}$$
 * $$\lambda = \sqrt{\alpha^2 - \beta^2}$$

Depending on the value of $$\lambda$$ the equation will have the following root

One Real Root
 * $$-\alpha = - \frac{B}{2A}$$

Two Real Roots
 * $$-\alpha \pm \lambda $$
 * $$-\frac{B}{2A} \pm \sqrt {\frac{B^2 - 4 AC}{2A}} $$

Two Complex Roots
 * $$-\alpha \pm j\lambda $$
 * $$-\frac{B}{2A} \pm j\sqrt {\frac{B^2 - 4 AC}{2A}} $$

Method 2
$$ax^2 + b = 0$$
 * $$x^2 + \frac{b}{a} = 0$$
 * $$x = \pm \sqrt$$
 * $$x = \pm j \sqrt{\frac{b}{a}}$$

Method 3
$$ax^2 - b = 0$$


 * $$x^2 - \frac{b}{a} = 0$$
 * $$x = \pm \sqrt{\frac{b}{a}}$$
 * $$x = \pm \sqrt{\frac{b}{a}}$$