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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) = ax+ 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

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.

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

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.the half of the coefficient of x i.e. on both sides

Transfer the constant on RHS then add square

Transfer the constant on RHS then add square

Step 4: Now write LHS as perfect square and simplify the RHS.

Now write LHS as perfect square and simplify the RHS

Step 5: Take the square root on both the sides.

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.

Now shift all the constant terms to the RHS

3. Quadratic formula method

In this method, we can find the roots by using quadratic formula. The quadratic formula is

Quadratic formula method

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 -

Nature of Roots

Where D = b2 – 4ac

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

so it is called the discriminant

∆ = Discriminant
Value of Discriminant

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