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We are already acquainted with quadratic equation and its roots. In this module, we shall study quadratic expression from the point of view of a function. It is a polynomial function of degree 2. The general form of quadratic expression/ function is :

$f\left(x\right)=a{x}^{2}+bx+c;\phantom{\rule{1em}{0ex}}a,b,c\in R,\phantom{\rule{1em}{0ex}}a>0$

$a{x}^{2}+bx+c=0;\phantom{\rule{1em}{0ex}}a,b,c\in R,\phantom{\rule{1em}{0ex}}a>0$

Nature of a given quadratic function is best understood in terms of discriminant, D, of corresponding quadratic equation. This is given as :

$D={b}^{2}-4ac$

Quadratic equation is obtained by equating quadratic function to zero. Quadratic equation has at most two roots. The roots are given by :

$\alpha =\frac{-b-\sqrt{D}}{2a}=\frac{-b-\sqrt{{b}^{2}-4ac}}{2a}$

$\beta =\frac{-b+\sqrt{D}}{2a}=\frac{-b+\sqrt{{b}^{2}-4ac}}{2a}$

## Properties of roots of quadratic equation

1 : If D>0, then roots are real and distinct.

2 : If D=0, then roots are real and equal.

3 : If D<0, then roots are complex conjugates with non-zero imaginary part.

4 : If D>0; a,b,c∈T (rational numbers) and D is a perfect square, then roots are rational.

5 : If D>0; a,b,c∈T (rational numbers) and D is not a perfect square, then roots are radical conjugates.

6 : If D>0; a=1;b,c∈Z (integer numbers) and roots are rational, then roots are integers.

7 : If a quadratic equation has more than two roots, then the function is an identity in x and a=b=c=0.

8 : If a quadratic equation has one real root and a,b,c∈R, then other root is also real.

The real roots of the quadratic equation are zeroes of quadratic function. The zeroes of quadratic function are real values of x for which value of quadratic function becomes zero. On graph, zeros are the points at which graph intersects y=0 i.e. x-axis.

Graph reveals important characteristics of quadratic function. The graph of quadratic function is a parabola. Working with the quadratic function, we have :

$y=a{x}^{2}+bx+c=a\left({x}^{2}+\frac{b}{a}x+\frac{c}{a}\right)$

In order to complete square, we add and subtract ${b}^{2}/4{a}^{2}$ as :

$⇒y=a\left({x}^{2}+\frac{b}{a}x+\frac{{b}^{2}}{4{a}^{2}}+\frac{c}{a}-\frac{{b}^{2}}{4{a}^{2}}\right)$

$⇒y=a\left\{{\left(x+\frac{b}{2a}\right)}^{2}-\frac{{b}^{2}-4ac}{4a}\right\}$

$⇒y+\frac{{b}^{2}-4ac}{4a}=a{\left(x+\frac{b}{2a}\right)}^{2}$

$⇒y+\frac{D}{4a}=a{\left(x+\frac{b}{2a}\right)}^{2}$

$⇒Y=a{X}^{2}$

Where,

$X=x+\frac{b}{2a}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}Y=y+\frac{D}{4a}$

Clearly, $Y=a{X}^{2}$ is an equation of parabola having its vertex given by (-b/2a, -D/4a). When a>0, parabola opens up and when a<0, parabola opens down. Further, parabola is symmetric about x=-b/2a.

## Maximum and minimum values of quadratic function

The graph of quadratic function extends on either sides of x-axis. Its domain, therefore, is R. On the other hand, value of function extends from vertex to either positive or negative infinity, depending on whether “a” is positive or negative.

When a>0, the graph of quadratic function is parabola opening up. The minimum and maximum values of the function are given by :

${y}_{\mathrm{min}}=-\frac{D}{4a}\phantom{\rule{1em}{0ex}}\text{at}\phantom{\rule{1em}{0ex}}x=-\frac{b}{2a}$

${y}_{\mathrm{max}}⇒\infty$

Clearly, range of the function is [-D/4a, ∞).

When a<0, the graph of quadratic function is parabola opening down. The maximum and minimum values of the function are given by :

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x

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