- Books Name
- Mathematics Book for CBSE Class 10

- Publication
- Carrier Point

- Course
- CBSE Class 10

- Subject
- Mathmatics

**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) = ax ^{2 }+ 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** ax^{2 }+ bx + c = 0, where a ≠ 0, b ≠ 0, c ≠ 0

**2. Pure Quadratic Equation** ax^{2 }= 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 ax^{2}+ bx + c = 0 such that aα^{2 }+ 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) =ax^{2}+bx+c is same as the roots of the Quadratic Equation ax^{2}+ 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 ax^{2 }+ 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)

ax^{2 }+ 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 }- b^{2 }= 0 to find the roots.

**Step1**: Given Quadratic Equation in the standard form ax^{2 }+ 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 b^{2 }– 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 = b ^{2} – 4ac**

The nature of the roots of the equation depends upon the value of D, so it is called the **discriminant.**

**∆ = Discriminant**