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Introduction

In grade 10, the basics of solving linear equations, quadratic equations, exponential equations and linear inequalities were studied. This chapter extends on that work. We look at different methods of solving quadratic equations.

Solution by factorisation

How to solve quadratic equations by factorisation was discussed in Grade 10. Here is an example to remind you of what is involved.

Solve the equation 2 x 2 - 5 x - 12 = 0 .

  1. This equation has no common factors.

  2. The equation is in the required form, with a = 2 , b = - 5 and c = - 12 .

  3. 2 x 2 - 5 x - 12 has factors of the form:

    ( 2 x + s ) ( x + v )

    with s and v constants to be determined. This multiplies out to

    2 x 2 + ( s + 2 v ) x + s v

    We see that s v = - 12 and s + 2 v = - 5 . This is a set of simultaneous equations in s and v , but it is easy to solve numerically. All the options for s and v are considered below.

    s v s + 2 v
    2 -6 -10
    -2 6 10
    3 -4 -5
    -3 4 5
    4 -3 -2
    -4 3 2
    6 -2 2
    -6 2 -2

    We see that the combination s = 3 and v = - 4 gives s + 2 v = - 5 .

  4. ( 2 x + 3 ) ( x - 4 ) = 0
  5. If two brackets are multiplied together and give 0, then one of the brackets must be 0, therefore

    2 x + 3 = 0

    or

    x - 4 = 0

    Therefore, x = - 3 2 or x = 4

  6. The solutions to 2 x 2 - 5 x - 12 = 0 are x = - 3 2 or x = 4 .

It is important to remember that a quadratic equation has to be in the form a x 2 + b x + c = 0 before one can solve it using the following methods.

Solve for a : a ( a - 3 ) = 10

  1. Remove the brackets and move all terms to one side.

    a 2 - 3 a - 10 = 0
  2. ( a + 2 ) ( a - 5 ) = 0
  3. a + 2 = 0

    or

    a - 5 = 0

    Solve the two linear equations and check the solutions in the original equation.

  4. Therefore, a = - 2 or a = 5

Solve for b : 3 b b + 2 + 1 = 4 b + 1

  1. 3 b ( b + 1 ) + ( b + 2 ) ( b + 1 ) ( b + 2 ) ( b + 1 ) = 4 ( b + 2 ) ( b + 2 ) ( b + 1 )
  2. The denominators are the same, therefore the numerators must be the same.

    However, b - 2 and b - 1

  3. 3 b 2 + 3 b + b 2 + 3 b + 2 = 4 b + 8 4 b 2 + 2 b - 6 = 0 2 b 2 + b - 3 = 0
  4. ( 2 b + 3 ) ( b - 1 ) = 0 2 b + 3 = 0 o r b - 1 = 0 b = - 3 2 o r b = 1
  5. Both solutions are valid

    Therefore, b = - 3 2 or b = 1

Solution by factorisation

Solve the following quadratic equations by factorisation. Some answers may be left in surd form.

  1. 2 y 2 - 61 = 101
  2. 2 y 2 - 10 = 0
  3. y 2 - 4 = 10
  4. 2 y 2 - 8 = 28
  5. 7 y 2 = 28
  6. y 2 + 28 = 100
  7. 7 y 2 + 14 y = 0
  8. 12 y 2 + 24 y + 12 = 0
  9. 16 y 2 - 400 = 0
  10. y 2 - 5 y + 6 = 0
  11. y 2 + 5 y - 36 = 0
  12. y 2 + 2 y = 8
  13. - y 2 - 11 y - 24 = 0
  14. 13 y - 42 = y 2
  15. y 2 + 9 y + 14 = 0
  16. y 2 - 5 k y + 4 k 2 = 0
  17. y ( 2 y + 1 ) = 15
  18. 5 y y - 2 + 3 y + 2 = - 6 y 2 - 2 y
  19. y - 2 y + 1 = 2 y + 1 y - 7

Questions & Answers

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
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
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
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
Do you know which machine is used to that process?
s.
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
Other chapter Q/A we can ask
Moahammedashifali Reply

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Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
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