Quadratic equation

 Quadratic equation 

•A polynomial equation of degree two is called a quadratic equation.

• General form of a quadratic equation (variable is x ) is: 

• A few examples of Quadratic equation are as follows: 

2x^2 +3x=0

x^2+1=0

3x^2=0.

• Solution of a quadratic equation : 

A quadratic equation can be solved in the following way: 

By factorising: 

Quadratic equation 

This formula is known as Shreedhar Acharya formula.

• Discriminant: 

The term : (b^2-4ac)  is known as discriminant of the quadratic equation. It determines the nature of roots of the Quadratic equation.

* If the discriminant of the quadratic equation is zero, the quadratic equation has real and equal roots. 

* If the discriminant is greater than zero , the quadratic equation has real and distinct roots.

* If the discriminant is less than zero, the quadratic equation has imaginary roots.

• Relation between roots and coefficients: 

If p and q are two roots of the Quadratic equation, then: 

p+q= (-b/a)       and    pq= (c/a).

That means: 

ap gp full form

 ap and gp full form:

ap gp full form

ap gp full form

In mathematics, more specifically in algebra ap and gp are very important topics.

 ap means Arithmetic progression or A.P.

And ,

gp means Geometric progression or G.P.

A.P. or ap or Arithmetic progression is a series of numbers: 1, 3, 5, ...  Or, 2,4,6,...

Whereas, 

G.P. or Geometric progression is a series of numbers: 1, 3,9,...   Or, 2,4,8,...

There are some differences between ap and gp.

ap , gp, ap gp full form

Arithmetic progression VS Geometric progression: ap vs gp

ap vs gp

Arithmetic progression VS Geometric progression:

1,2,3,...    VS   3,9,27,...

Let's Closely and carefully watch both series...🤫

 What did we see?

In the first series: 1,2,3...

  1, 1+1, 2+1,...and so on...

That means, adding 1 to the previous term of the series we get the next term.

Also, the series starts with 1 and two consecutive terms has a common difference of 1.

This type of series is known as Arithmetic progression (ap or, A.P.)

Now, let's go to the second series: 3,9,27,...

 3, 3×3, 9×3,...and so on...

Here, if we multiply the previous term with 3, we get the next term.

 The series starts with 3 and two consecutive terms has a common ratio 3.

This type of series is known as Geometric progression (gp or, G.P.)

 

The following image clearly and briefly reflects ap vs gp. Difference and relation between Arithmetic progression and Geometric progression: ap vs gp.

i.e., in a more general mathematical form:

 ap or A.P. or Arithmetic progression looks like:  

                      a, a+d, a+2d,...

a= first term, d= common difference.

And, gp or G.P. or Geometric progression looks like:

                       a, ar, ar×r,...

a= first term, r= common ratio.

 In brief, ap vs gp :

 ap or Arithmetic progression is related to addition of terms with a fixed number.

And, gp or Geometric progression is related to multiplication of terms with a fixed number.

Thanks.