Arithmetic Course/Polynominal Equation

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Polynominal Equation[edit]

An equation is an expression of one variable such that

f(x) = Ax^n + Bx^(n-1) + x^1 + x^0 = 0 .

Solving Polynominal Equation[edit]

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

First Order Equation[edit]

A first order polynominal 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[edit]

A second order polynominal 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

Solving Equation[edit]

Method 1[edit]

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[edit]

ax^2 + b = 0

x^2 + \frac{b}{a} = 0
x = \pm \sqrt{-\frac{b}{a}}
x = \pm j \sqrt{\frac{b}{a}}

Method 3[edit]

ax^2 - b = 0

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