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## 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.

$\begin{array}{ccc}\hfill 2{x}^{2}+2x& =& 1\hfill \\ \hfill \frac{2-x}{3x+1}& =& 2x\hfill \\ \hfill \frac{4}{3}x-6& =& 7{x}^{2}+2\hfill \end{array}$

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:

$\left(x+1\right)\left(2x-3\right)=2{x}^{2}-x-3.$

In order to solve:

$2{x}^{2}-x-3=0$

we need to be able to write $2{x}^{2}-x-3$ as $\left(x+1\right)\left(2x-3\right)$ , 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 $f\left(x\right)g\left(x\right)=0$ , for some functions $f$ and $g$ , we know that the solution is $f\left(x\right)=0$ OR $g\left(x\right)=0$ , 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

1. $x+{x}^{2}$
2. ${x}^{2}+1+2x$
3. ${x}^{2}-4x+5$
4. $16{x}^{2}-9$
5. $4{x}^{2}+4x+1$

Being able to factorise a quadratic means that you are one step away from solving a quadratic equation. For example, ${x}^{2}-3x-2=0$ can be written as $\left(x-1\right)\left(x-2\right)=0$ . 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 ${x}^{2}-3x-2=0$ .

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

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

Solve for $x$ : $3{x}^{2}+2x-1=0$

1. As we have seen the factors of $3{x}^{2}+2x-1$ are $\left(x+1\right)$ and $\left(3x-1\right)$ .

2. $\left(x+1\right)\left(3x-1\right)=0$
3. We have

$x+1=0$

or

$3x-1=0$

Therefore, $x=-1$ or $x=\frac{1}{3}$ .

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

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,
$\begin{array}{ccc}\hfill ax+b& =& \frac{c}{x}\hfill \\ \hfill x\left(ax+b\right)& =& x\left(\frac{c}{x}\right)\hfill \\ \hfill a{x}^{2}+bx& =& c\hfill \end{array}$
• This is raising both sides to the power of $-1$ . For example,
$\begin{array}{ccc}\hfill \frac{1}{a{x}^{2}+bx}& =& c\hfill \\ \hfill {\left(\frac{1}{a{x}^{2}+bx}\right)}^{-1}& =& {\left(c\right)}^{-1}\hfill \\ \hfill \frac{a{x}^{2}+bx}{1}& =& \frac{1}{c}\hfill \\ \hfill a{x}^{2}+bx& =& \frac{1}{c}\hfill \end{array}$
• This is raising both sides to the power of 2. For example,
$\begin{array}{ccc}\hfill \sqrt{a{x}^{2}+bx}& =& c\hfill \\ \hfill {\left(\sqrt{a{x}^{2}+bx}\right)}^{2}& =& {c}^{2}\hfill \\ \hfill a{x}^{2}+bx& =& {c}^{2}\hfill \end{array}$

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
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
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
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
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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 a peer
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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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