ax

^{2}+ bx + c,

where ‘a’, ‘b’ and ‘c’ are the coefficient of quadratic polynomial expression. They are also constant where ‘a’ should never be '0'. In equation ‘x’ is a variable. A quadratic polynomial is a polynomial of degree 2. Now we will derive the quadratic polynomial to find the roots of polynomial and for that we divide the above equation by co-efficient ‘x’ so the equation is,

x

^{2}+ b/a + c/a=0,

After this we will transfer the constant term of polynomial on the right hand side,

x

^{2}+ b/a = -c/a,

By using the binomial formula,

x

^{2}+ 2.b/2a + (b/2a)

^{2}= (x+b/2a)

^{2},

To complete the squares we add (b/2a)

^{2}, on both sides of our equation:

x

^{2}+ (b/a).x + (b/2a)

^{2}= -c/a + (b/2a)

^{2},

Now we have completed the Square of left hand side so we simplify the right hand side,

(x+b / 2a)

^{2}= b

^{2}-4ac/4a

^{2},

Taking square roots on both sides we get,

(x + b / 2a)=+√(b

^{2}- 4ac)/4a

^{2}or (x + b/2a) = -√(b

^{2}- 4ac)/4a

^{2},

We get two roots because under root contain both positive and negative feature with it. So the Quadratic Equation contains two roots according to nature of sign which is

x = -(b + √b

^{2-}4ac)/2a and x=-(b - √b

^{2}-4ac)/2a,

Let us take the example to understand better. We have a quadratic polynomial,

x

^{2}+ 2x - 7 in this values of a, b and c are respectively 1, 2 and -7. After using that Quadratic Formula

x = (-2+√2

^{2}-(4).( 1). (-7))/2 * 1 and x = (-2-√2

^{2}-4 (1). (-7))/2 * 1,

x=-1+2√2 and x=-1-2√2,

Quadratic polynomial is degree of two, that’s why it is also called second order polynomial. In quadratic polynomial coefficient 'a' may be real and complex. In the quadratic polynomial the sum of products of roots is equal to ‘b/a’ which is generally written as ‘-p’ and the product of roots of quadratic polynomial is ‘-c/a’ which is also written as ‘q’, so with the help of sum of roots and sum of products we can write the quadratic equation in following manner:-

x

^{2}- p*x + q,

where ‘p’ is sum of roots equals to -b/a, and ‘q’ is products of roots equals to c/a.