Equation of the form is known as quadratic equation in one variable and forms a second degree polynomial, where;
 a , b, c are the real numbers
 a ≠ 0
Consider an equation ax^{2}+bx+c=0 then we can further define
Complete quadratic equation: where both the coefficients and the constant term are nonzero. a≠ 0, b≠ 0, c≠ 0.
Example
2x^{2}+ 3x+ 4= 0
Pure quadratic equation: where except the coefficient of x^{2}, others can be zero. a≠ 0 , b=0, c=0
Examples
2x^{2}= 0
2x^{2}+3=0
ZERO OF A QUADRATIC EQUATION
It is the value of x for which the quadratic equation becomes zero.
A quadratic equation (the two degree polynomial) will have at most 2 zeros and hence at most 2 roots.
NOTE: Zeros and roots of a second degree polynomial are the same.
If the equation is given by ax^{2}+bx+c, thus if then the roots of quadratic equation are given by the quadratic formula given as,
METHODS TO SOLVE A QUADRATIC EQUATIONS

Factorizing
By Splitting the midterm
Example: Consider an equation x^{2} + 5x + 6= 0
This equation can be written as
x^{2}+3x+2x+6= 0
x(x+3)+2(x+3)= 0
(x+2)(x+3)= 0
Hence the roots are x= 2 or x= 3

Completing the squares
The equation is converted in the form a^{2}+b^{2} or a^{2}b^{2} by subtracting or adding a suitable constant.
Example: Consider an equation 4x^{2} – 2x – 5 = 0
This equation can be written as
While solving further we want the x^{2 }term by itself, therefore we have to divide the whole equation by whatever is multiplied by it.
In this case divide the whole equation by 4 to get
Now, we are going to work with the x term along with its sign, here take as the numeric coefficient here.
Multiply to this numeric coefficient to get the derived coefficient
Square the derived coefficient to get
Get back to the equation
Add to both sides of the equation
By doing this we have created the left hand side which is a perfect square and hence can be converted into a squared binomial.
The above equation is in complete square form.
To get the roots, apply square root to both the sides.
Hence, the roots of the equation are as follows

Quadratic Formula
Find the determinant and use the quadratic formula to take out the roots.
Consider the above example itself 4x^{2} – 2x – 5 = 0
The quadratic formula, as mentioned previously is:

Discriminant method
 Find
 Follow the decision rule:
VALUE OF D  ROOTS  NATURE OF ROOTS 
D>0  –b+ /2a and –b /2a  Real and distinct 
D=0  b/2a  Real and equal OR coincidental 
D<0  no real roots  Not Real (imaginary roots) 
Sum and product of roots
Let α and β be the two distinct roots of the equation , then
 Sum of roots is given as
= α + β =
 Product of roots is given as
= α.β =
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