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In grade 10, you learnt how to solve sets of simultaneous equations where both equations were linear (i.e. had the highest power equal to 1). In this chapter, you will learn how to solve sets of simultaneous equations where one is linear and one is quadratic. As in Grade 10, the solution will be found both algebraically and graphically.

The only difference between a system of linear simultaneous equations and a system of simultaneous equations with one linear and one quadratic equation, is that the second system will have at most two solutions.

An example of a system of simultaneous equations with one linear equation and one quadratic equation is:

y - 2 x = - 4 x 2 + y = 4

Graphical solution

The method of graphically finding the solution to one linear and one quadratic equation is identical to systems of linear simultaneous equations.

Method: graphical solution to a system of simultaneous equations with one linear and one quadratic equation

  1. Make y the subject of each equation.
  2. Draw the graphs of each equation as defined above.
  3. The solution of the set of simultaneous equations is given by the intersection points of the two graphs.

For this example, making y the subject of each equation, gives:

y = 2 x - 4 y = 4 - x 2

Plotting the graph of each equation, gives a straight line for the first equation and a parabola for the second equation.

The parabola and the straight line intersect at two points: (2,0) and (-4,-12). Therefore, the solutions to the system of equations in [link] is x = 2 , y = 0 and x = - 4 , y = 12

Solve graphically:

y - x 2 + 9 = 0 y + 3 x - 9 = 0
  1. For the first equation:

    y - x 2 + 9 = 0 y = x 2 - 9

    and for the second equation:

    y + 3 x - 9 = 0 y = - 3 x + 9
  2. The graphs intersect at ( - 6 , 27 ) and at ( 3 , 0 ) .

  3. The first solution is x = - 6 and y = 27 . The second solution is x = 3 and y = 0 .

Graphical solution

Solve the following systems of equations graphically. Leave your answer in surd form, where appropriate.

  1. b 2 - 1 - a = 0 , a + b - 5 = 0
  2. x + y - 10 = 0 , x 2 - 2 - y = 0
  3. 6 - 4 x - y = 0 , 12 - 2 x 2 - y = 0
  4. x + 2 y - 14 = 0 , x 2 + 2 - y = 0
  5. 2 x + 1 - y = 0 , 25 - 3 x - x 2 - y = 0

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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