Finding Roots of a Quadratic Polynomial Equation

1. A Quadratic Polynomial Equation in 1 Variable is given as follows
   
   \(Ax^2 + Bx + C =0\)   ...(1)
   
   where the Co-efficient of \(x^2\) Cannot be Zero i.e. \(A\neq0\)
2. Any Quadratic Polynomial Equation in 1 Variable always has 2 Roots/Solutions given by the formulae
   
   \(x= {\Large \frac{-B\hspace{1mm}+\hspace{1mm}D_{R1}}{2A}}, \hspace{.6cm} x= {\Large \frac{-B\hspace{1mm}+\hspace{1mm}D_{R2}}{2A}} \)   ...(2)
   
   where \(D_{R1}\) and \(D_{R2}\) are the 2 Square Roots of the Discriminant \(D\) calculated as
   
   \(D=B^2-4AC\)   ...(3)
   
   Since the 2 Square Roots of the Discriminant only differ in sign (i.e. One Square Root is Negative of the Other), the 2 Square Roots of the Discriminant can be specified by a common variable \(D_{R}\) and accordingly, the formula for calculation of Roots/Solutions of Quadratic Equation is given as
   
   \(x = {\Large \frac{-B \hspace{1mm}\pm \hspace{1mm} D_{R}}{2A}}\hspace{5mm}\Rightarrow x = {\Large \frac{-B\hspace{1mm}\pm\hspace{1mm}\sqrt{B^2-4AC}}{2A}}\)   ...(4)
   
   
3. Following are the Steps for Deriving the formula for Finding Roots of Quadratic Polynomial Equation
   1. Take the Constant of the equation (1) to the other side as given in the following
      
      \(Ax^2 + Bx = -C\)
   2. Divide the equation with Co-efficient of \(x^2\) (i.e. \(A\)) on both sides
      
      \(x^2 + {\Large \frac{B}{A}}x = {\Large \frac{-C}{A}}\)
   3. Divide Co-efficient of \(x\) by 2 and Add the Square of Result (i.e. \({\Large{(\frac{B}{2A})}^2}\)) on both sides
      
      \(x^2 + {\Large \frac{B}{A}}x + {\Large \frac{B^2}{4A^2}}={\Large \frac{-C}{A}} + {\Large \frac{B^2}{4A^2}}\)
   4. Since the Left Hand Side of the equation is a Complete Square, the equation can be written as
      
      \({(x + {\Large \frac{B}{2A}})}^2={\Large \frac{B^2-4AC}{4A^2}}\)
      
      \(\Rightarrow x + {\Large \frac{B}{2A}}= {\Large \frac{\pm\sqrt{B^2-4AC}}{2A}}\)
      
      \(\Rightarrow x = {\Large \frac{-B\hspace{1mm}\pm\hspace{1mm}\sqrt{B^2-4AC}}{2A}}\)
      
      which is the Formula for Finding Roots/Solutions of Quadratic Polynomial Equations.
4. The 2 Roots/Solutions of the Quadratic Polynomial Equation in 1 Variable can be of following 6 types depending on the value of \(D\) as follows
   1. If \(D\) is Real and \(D>0\) then Both Roots are Real and Distinct
   2. If \(D=0\) then Both Roots are Same (either Real or Complex)
   3. If \(D\) is Real and \(D<0\) then 2 Roots are Conjugate Complex.
   4. If \(D\) is Complex then Both Roots are Distinct and either Both are Complex or One is Real and One is Complex.
   You can use the Quadratic Equation Roots Calculator to find the Roots/Solutions of Quadratic Polynomial Equations.