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Maths grade 10 rought draft
Equations and inequalities
Solving quadratic equations
You can combine these in many ways and so the best way to develop your intuition for the best thing to do is practice problems. A combined set of operations could be, for example,
$$\begin{array}{cccc}\hfill \frac{1}{\sqrt{a{x}^{2}+bx}}& =& c\hfill & \\ \hfill {\left(\frac{1}{a{x}^{2}+bx}\right)}^{-1}& =& {\left(c\right)}^{-1}\hfill & \left(\mathrm{invert\; both\; sides}\right)\hfill \\ \hfill \frac{\sqrt{a{x}^{2}+bx}}{1}& =& \frac{1}{c}\hfill & \\ \hfill \sqrt{a{x}^{2}+bx}& =& \frac{1}{c}\hfill & \\ \hfill {\left(\sqrt{a{x}^{2}+bx}\right)}^{2}& =& {\left(\frac{1}{c}\right)}^{2}\hfill & \left(\mathrm{square\; both\; sides}\right)\hfill \\ \hfill a{x}^{2}+bx& =& \frac{1}{{c}^{2}}\hfill & \end{array}$$
Solve for
$x$ :
$\sqrt{x+2}=x$
Square both sides of the equation
Both sides of the equation should be squared to remove the square root sign.
$$x+2={x}^{2}$$
Write equation in the form
$a{x}^{2}+bx+c=0$
$$\begin{array}{cccc}\hfill x+2& =& {x}^{2}\hfill & (\mathrm{subtract}\phantom{\rule{2pt}{0ex}}{\mathrm{x}}^{2}\phantom{\rule{2pt}{0ex}}\mathrm{from\; both\; sides})\hfill \\ \hfill x+2-{x}^{2}& =& 0\hfill & (\mathrm{divide\; both\; sides\; by}\phantom{\rule{2pt}{0ex}}-1)\hfill \\ \hfill -x-2+{x}^{2}& =& 0\hfill & \\ \hfill {x}^{2}-x+2& =& 0\hfill & \end{array}$$
Factorise the quadratic
$${x}^{2}-x+2$$
The factors of
${x}^{2}-x+2$ are
$(x-2)(x+1)$ .
Write the equation with the factors
$$(x-2)(x+1)=0$$
Determine the two solutions
We have
$$x+1=0$$
or
$$x-2=0$$
Therefore,
$x=-1$ or
$x=2$ .
Check whether solutions are valid
Substitute
$x=-1$
into the original equation
$\sqrt{x+2}=x$ :
$$\begin{array}{ccc}\hfill \mathrm{LHS}& =& \sqrt{(-1)+2}\hfill \\ & =& \sqrt{1}\hfill \\ & =& 1\hfill \\ \hfill \mathrm{but}\\ \hfill \mathrm{RHS}& =& (-1)\hfill \end{array}$$
Therefore LHS
$\xe2\u2030$ RHS. The sides of an equation must always balance, a potential solution that does not balance the equation is not valid. In this case the equation does not balance.
Therefore
$x\xe2\u2030-1$ .
Now substitute
$x=2$ into original equation
$\sqrt{x+2}=x$ :
$$\begin{array}{ccc}\hfill \mathrm{LHS}& =& \sqrt{2+2}\hfill \\ & =& \sqrt{4}\hfill \\ & =& 2\hfill \\ \hfill \mathrm{and}\\ \hfill \mathrm{RHS}& =& 2\hfill \end{array}$$
Therefore LHS = RHS
Therefore
$x=2$ is the only valid solution
Write the final answer
$\sqrt{x+2}=x$ for
$x=2$ only.
Solve the equation:
${x}^{2}+3x-4=0$ .
Check if the equation is in the form
$a{x}^{2}+bx+c=0$
The equation is in the required form, with
$a=1$ .
Factorise the quadratic
You need the factors of 1 and 4 so that the middle term is
$+3$ So the factors are:
$(x-1)(x+4)$
Solve the quadratic equation
$${x}^{2}+3x-4=(x-1)(x+4)=0$$
Therefore
$x=1$ or
$x=-4$ .
Check the solutions
${1}^{2}+3\left(1\right)-4=0$
${(-4)}^{2}+3(-4)-4=0$
Both solutions are valid.
Write the final solution
Therefore the solutions are
$x=1$ or
$x=-4$ .
Find the roots of the quadratic
equation
$0=-2{x}^{2}+4x-2$ .
Determine whether the equation is in the form
$a{x}^{2}+bx+c=0$ , with no
common factors.
There is a common factor: -2.
Therefore, divide both sides of the equation by -2.
$$\begin{array}{ccc}\hfill -2{x}^{2}+4x-2& =& 0\hfill \\ \hfill {x}^{2}-2x+1& =& 0\hfill \end{array}$$
Factorise
${x}^{2}-2x+1$
The middle term is negative. Therefore, the factors are
$(x-1)(x-1)$
If we multiply out
$(x-1)(x-1)$ , we get
${x}^{2}-2x+1$ .
Solve the quadratic equation
$${x}^{2}-2x+1=(x-1)(x-1)=0$$
In this case, the quadratic is a perfect square, so there is only one solution
for
$x$ :
$x=1$ .
Check the solution
$-2{\left(1\right)}^{2}+4\left(1\right)-2=0$ .
Write the final solution
The root of
$0=-2{x}^{2}+4x-2$ is
$x=1$ .
Solving quadratic equations
Solve for
$x$ :
$(3x+2)(3x-4)=0$
Solve for
$x$ :
$(5x-9)(x+6)=0$
Solve for
$x$ :
$(2x+3)(2x-3)=0$
Solve for
$x$ :
$(2x+1)(2x-9)=0$
Solve for
$x$ :
$(2x-3)(2x-3)=0$
Solve for
$x$ :
$20x+25{x}^{2}=0$
Solve for
$x$ :
$4{x}^{2}-17x-77=0$
Solve for
$x$ :
$2{x}^{2}-5x-12=0$
Solve for
$x$ :
$-75{x}^{2}+290x-240=0$
Solve for
$x$ :
$2x=\frac{1}{3}{x}^{2}-3x+14\frac{2}{3}$
Solve for
$x$ :
${x}^{2}-4x=-4$
Solve for
$x$ :
$-{x}^{2}+4x-6=4{x}^{2}-5x+3$
Solve for
$x$ :
${x}^{2}=3x$
Solve for
$x$ :
$3{x}^{2}+10x-25=0$
Solve for
$x$ :
${x}^{2}-x+3$
Solve for
$x$ :
${x}^{2}-4x+4=0$
Solve for
$x$ :
${x}^{2}-6x=7$
Solve for
$x$ :
$14{x}^{2}+5x=6$
Solve for
$x$ :
$2{x}^{2}-2x=12$
Solve for
$x$ :
$3{x}^{2}+2y-6={x}^{2}-x+2$
Questions & Answers
how can chip be made from sand
is this allso about nanoscale material
Almas
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
where is the latest information on a no technology how can I find it
William
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
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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 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
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:
OpenStax, Maths grade 10 rought draft. OpenStax CNX. Sep 29, 2011 Download for free at http://cnx.org/content/col11363/1.1
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