Quadratic Equations
Quadratic Polynomial
A polynomial whose degree is 2, is called a quadratic polynomial. It is in the form of
p(x) = ax2 + bx + c, where a ≠ 0
Quadratic Equation
When we equate the quadratic polynomial to zero then it is called a Quadratic Equation i.e. if
p(x) = 0, then it is known as Quadratic Equation.
Standard form of Quadratic Equation
ax2 + bx + c = 0
where a, b, c are the real numbers and a≠0
Types of Quadratic Equations
1. Complete Quadratic Equation ax2 + bx + c = 0, where a ≠ 0, b ≠ 0, c ≠ 0
2. Pure Quadratic Equation ax2 = 0, where a ≠ 0, b = 0, c = 0
Roots of a Quadratic Equation
Let x = α where α is a real number. If α satisfies the Quadratic Equation ax2+ bx + c = 0 such that aα2 + bα + c = 0, then α is the root of the Quadratic Equation.
As quadratic polynomials have degree 2, therefore Quadratic Equations can have two roots. So the zeros of quadratic polynomial p(x) =ax2+bx+c is the same as the roots of the Quadratic Equation ax2+ bx + c= 0.
Methods to solve the Quadratic Equations
There are three methods to solve Quadratic Equations-
1. Factorisation Method
In this method, we factorise the equation into two linear factors and equate each factor to zero to find the roots of the given equation.
Step 1: Given Quadratic Equation in the form of ax2 + bx + c = 0.
Step 2: Split the middle term bx as mx + nx so that the sum of m and n is equal to b and the product of m and n is equal to ac.
Step3: By factorization we get the two linear factors (x + p) and (x + q)
ax2 + bx + c = 0 = (x + p) (x + q) = 0
Step 4: Now we have to equate each factor to zero to find the value of x.
These values of x are the two roots of the given Quadratic Equation.
2. Completing the square method
In this method, we convert the equation in the square form (x + a)2 - b2 = 0 to find the roots.
Step1: Given Quadratic Equation in the standard form ax2 + bx + c = 0.
Step 2: Divide both sides by a
Step 3: Transfer the constant on RHS then add square of the half of the coefficient of x i.e.
on both sides
Step 4: Now write LHS as perfect square and simplify the RHS.
Step 5: Take the square root on both sides.
Step 6: Now shift all the constant terms to the RHS and we can calculate the value of x as there is no variable at the RHS.
3. Quadratic formula method
In this method, we can find the roots by using the quadratic formula. The quadratic formula is
where a, b and c are the real numbers and b2 – 4ac is called discriminant.
To find the roots of the equation, put the value of a, b and c in the quadratic formula.
Nature of Roots
From the quadratic formula, we can see that the two roots of the Quadratic Equation are -
Where D = b2 – 4ac
The nature of the roots of the equation depends upon the value of D, so it is called the discriminant.
∆ = Discriminant