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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 x = a x 2 + b x + c ; a , b , c R , a > 0

Elements of quadratic equation

Quadratic equation

Quadratic equation is obtained by equating quadratic function to zero. General form of quadratic equation corresponding to quadratic function is :

a x 2 + b x + c = 0 ; a , b , c R , a > 0

Discriminant of quadratic equation

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 4 a c

Roots of quadratic equation

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

α = - b D 2 a = - b b 2 4 a c 2 a

β = - b + D 2 a = - b + b 2 4 a c 2 a

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.

Elements of quadratic function

Zeroes of quadratic function

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 of quadratic function

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 + b x + c = a x 2 + b a x + c a

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

y = a x 2 + b a x + b 2 4 a 2 + c a b 2 4 a 2

y = a { x + b 2 a 2 - b 2 4 a c 4 a }

y + b 2 4 a c 4 a = a x + b 2 a 2

y + D 4 a = a x + b 2 a 2

Y = a X 2

Where,

X = x + b 2 a and Y = y + D 4 a

Graph of quadratic function

The graph is parabola.

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 min = - D 4 a at x = - b 2 a

y max

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 :

Questions & Answers

How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply

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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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