<< Chapter < Page Chapter >> Page >

Equations and inequalities: solving quadratic equations

A quadratic equation is an equation where the power of the variable is at most 2. The following are examples of quadratic equations.

2 x 2 + 2 x = 1 2 - x 3 x + 1 = 2 x 4 3 x - 6 = 7 x 2 + 2

Quadratic equations differ from linear equations by the fact that a linear equation only has one solution, while a quadratic equation has at most two solutions. However, there are some special situations when a quadratic equation only has one solution.

We solve quadratic equations by factorisation, that is writing the quadratic as a product of two expressions in brackets. For example, we know that:

( x + 1 ) ( 2 x - 3 ) = 2 x 2 - x - 3 .

In order to solve:

2 x 2 - x - 3 = 0

we need to be able to write 2 x 2 - x - 3 as ( x + 1 ) ( 2 x - 3 ) , which we already know how to do. The reason for equating to zero and factoring is that if we attempt to solve it in a 'normal' way, we may miss one of the solutions. On the other hand, if we have the (non-linear) equation f ( x ) g ( x ) = 0 , for some functions f and g , we know that the solution is f ( x ) = 0 OR g ( x ) = 0 , which allows us to find BOTH solutions (or know that there is only one solution if it turns out that f = g ).

Investigation : factorising a quadratic

Factorise the following quadratic expressions:

  1. x + x 2
  2. x 2 + 1 + 2 x
  3. x 2 - 4 x + 5
  4. 16 x 2 - 9
  5. 4 x 2 + 4 x + 1

Being able to factorise a quadratic means that you are one step away from solving a quadratic equation. For example, x 2 - 3 x - 2 = 0 can be written as ( x - 1 ) ( x - 2 ) = 0 . This means that both x - 1 = 0 and x - 2 = 0 , which gives x = 1 and x = 2 as the two solutions to the quadratic equation x 2 - 3 x - 2 = 0 .

Method: solving quadratic equations

  1. First divide the entire equation by any common factor of the coefficients, so as to obtain an equation of the form a x 2 + b x + c = 0 where a , b and c have no common factors. For example, 2 x 2 + 4 x + 2 = 0 can be written as x 2 + 2 x + 1 = 0 by dividing by 2.
  2. Write a x 2 + b x + c in terms of its factors ( r x + s ) ( u x + v ) . This means ( r x + s ) ( u x + v ) = 0 .
  3. Once writing the equation in the form ( r x + s ) ( u x + v ) = 0 , it then follows that the two solutions are x = - s r or x = - u v .
  4. For each solution substitute the value into the original equation to check whether it is valid

Solutions of quadratic equations

There are two solutions to a quadratic equation, because any one of the values can solve the equation.

Khan academy video on equations - 3

Solve for x : 3 x 2 + 2 x - 1 = 0

  1. As we have seen the factors of 3 x 2 + 2 x - 1 are ( x + 1 ) and ( 3 x - 1 ) .

  2. ( x + 1 ) ( 3 x - 1 ) = 0
  3. We have

    x + 1 = 0

    or

    3 x - 1 = 0

    Therefore, x = - 1 or x = 1 3 .

  4. We substitute the answers back into the original equation and for both answers we find that the equation is true.
  5. 3 x 2 + 2 x - 1 = 0 for x = - 1 or x = 1 3 .

Got questions? Get instant answers now!

Sometimes an equation might not look like a quadratic at first glance but turns into one with a simple operation or two. Remember that you have to do the same operation on both sides of the equation for it to remain true.

You might need to do one (or a combination) of:

  • For example,
    a x + b = c x x ( a x + b ) = x ( c x ) a x 2 + b x = c
  • This is raising both sides to the power of - 1 . For example,
    1 a x 2 + b x = c ( 1 a x 2 + b x ) - 1 = ( c ) - 1 a x 2 + b x 1 = 1 c a x 2 + b x = 1 c
  • This is raising both sides to the power of 2. For example,
    a x 2 + b x = c ( a x 2 + b x ) 2 = c 2 a x 2 + b x = c 2

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Siyavula textbooks: grade 10 maths [caps]' conversation and receive update notifications?

Ask