Quadratic Polynomial : a polynomial in the form ax2 + bx + c where a,b,c are real nos. and a is not equal to 0 is called a quadratic polynomial. 

Zeroes of Quadratic Polynomial : The values of the variable as such when substituted, make the LHS value 0 are called zeroes of the polynomial. e.g. -2 and -3 are zeroes of polynomial x2 + 5x + 6. 

Roots of Quadratic Equation : The values of the variable which satisfy the given quadratic equation are called the roots of the quadratic equation. A quadratic equation always has 2 roots. 

Methods of solving Quadratic Equations :
Factorization Method : In this method, product of 1st and 3rd term is taken, and factorized in such a way that its factors are the addition or subtraction of the middle term. Then terms are grouped into twos which are factors of the equation and the inverses of coefficients are the roots of the quadratic equation.
Completing Square Method : In this method,the third term is transferred to the RHS. The LHS is made a perfect square using formula '3rd term = (1/2 coefficient of x)2'. The new third term is added to both sides and then using factorization method roots are calculated.
Formula Method : In this method, a formula is used. The answers are often inverses of one another. The formula is 
x = [-b + square root of(b2 - 4ac)] / 2a. 

Discriminant : When roots are found out by using formula method, the term b2 - 4ac is called the discriminant. It also determines the nature of the roots. 

Relation between roots and coefficients : If p and q are two roots of ax2 + bx + c = 0 then p + q = -b/a and pq = c/a. 

Forming Quadratic Equation from roots : If p and q are two roots, then the Quadratic Equation is x2 - (p + q)x + pq = 0. 

Reducible Equations : Sometimes, a given equation may not be quadratic, but it can be reduced to one. e.g. 5x4 - 22x2 + 8 = 0. Taking x2 = m, we get the equation 5m2 - 22m + 8 = 0.