Quadratic Equation Class 10 Notes: Chapter 4
Get the complete concepts covered in quadratic equations for class 10 Maths here. In this article, you will learn the concept of quadratic equations, standard form, nature of roots, methods for finding the solution for the given quadratic equations with more examples.
Revision Notes on 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
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 same as the roots of the Quadratic Equation ax2+ bx + c= 0.
Methods to solve the Quadratic Equations
There are three methods to solve the 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.
Step 3: 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 the 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 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