Page 1 / 1
This module contains methods on solving radical equations.

When solving equations that involve radicals , begin by asking yourself: is there an $x$ under the square root? The answer to this question will determine the way you approach the problem.

If there is not an $x$ under the square root—if only numbers are under the radicals—you can solve much the same way you would solve with no radicals at all.

## Radical equation with no variables under square roots

 $\sqrt{2}x+5=7–\sqrt{3}x$ Sample problem: no variables under radicals $\sqrt{2}+\sqrt{3}x=7-5$ Get everything with an $x$ on one side, everything else on the other $x\left(\sqrt{2}+\sqrt{3}\right)=2$ Factor out the $x$ $x=\frac{2}{\sqrt{2}+\sqrt{3}}$ Divide, to solve for $x$

The key thing to note about such problems is that you do not have to square both sides of the equation . $\sqrt{2}$ may look ugly, but it is just a number—you could find it on your calculator if you wanted to—it functions in the equation just the way that the number 10, or $\frac{1}{3}$ , or π would.

If there is an $x$ under the square root, the problem is completely different. You will have to square both sides to get rid of the radical. However, there are two important notes about this kind of problem.

1. Always get the radical alone, on one side of the equation , before squaring.
2. Squaring both sides can introduce false answers —so it is important to check your answers after solving!

Both of these principles are demonstrated in the following example.

## Radical equation with variables under square roots

 $\sqrt{x+2}+\mathrm{3x}=\mathrm{5x}+1$ Sample problem with variables under radicals $\sqrt{x+2}=\mathrm{2x}+1$ Isolate the radical before squaring! $x+2={\left(\mathrm{2x}+1\right)}^{2}$ Now, square both sides $x+2={\mathrm{4x}}^{2}+\mathrm{4x}+1$ Multiply out. Hey, it looks like a quadratic equation now! $x+2={\mathrm{4x}}^{2}+\mathrm{4x}+1$ As always with quadratics, get everything on one side. $\left(\mathrm{4x}-1\right)\left(x+1\right)=0$ Factoring: the easiest way to solve quadratic equations. $x=\frac{1}{4}$ or $x=-1$ Two solutions . Do they work? Check in the original equation !
 Check $x=\frac{1}{4}$ Check $x=–1$ $\sqrt{\frac{1}{4}+2}+3\left(\frac{1}{4}\right)\stackrel{?}{=}5\left(\frac{1}{4}\right)+1$ $\sqrt{-1+2}+3\left(-1\right)\stackrel{?}{=}5\left(-1\right)+1$ $\sqrt{\frac{1}{4}+\frac{8}{4}}+\frac{3}{4}\stackrel{?}{=}\frac{5}{4}+1$ $\sqrt{1}-3\stackrel{?}{=}-5+1$ $\sqrt{\frac{9}{4}}+\frac{3}{4}\stackrel{?}{=}\frac{5}{4}+\frac{4}{4}$ $1-3\stackrel{?}{=}-5+1$ $\frac{3}{2}+\frac{3}{4}\stackrel{?}{=}\frac{5}{4}+\frac{4}{4}$ $-2=-4$  Not equal! $\frac{9}{4}=\frac{9}{4}$

So the algebra yielded two solutions: $\frac{1}{4}$ and –1. Checking, however, we discover that only the first solution is valid. This problem demonstrates how important it is to check solutions whenever squaring both sides of an equation.

If variables under the radical occur more than once, you will have to go through this procedure multiple times. Each time, you isolate a radical and then square both sides.

## Radical equation with variables under square roots multiple times

 $\sqrt{x+7}-x=1$ Sample problem with variables under radicals multiple times $\sqrt{x+7}=\sqrt{x}+1$ Isolate one radical. (I usually prefer to start with the bigger one.) $x+7=x+2\sqrt{x}+1$ Square both sides. The two -radical equation is now a one -radical equation. $6=2\sqrt{x}$ $3=x$ Isolate the remaining radical, then square both sides again.. $9=x$ In this case, we end up with only one solution. But we still need to check it.
 $\sqrt{9+7}-\sqrt{9}\stackrel{?}{=}1$ $\sqrt{16}-\sqrt{9}\stackrel{?}{=}1$ $4-3=1$

Remember, the key to this problem was recognizing that variables under the radical occurred in the original problem two times . That cued us that we would have to go through the process—isolate a radical, then square both sides—twice, before we could solve for $x$ . And whenever you square both sides of the equation, it’s vital to check your answer(s)!

Why is it that—when squaring both sides of an equation—perfectly good algebra can lead to invalid solutions? The answer is in the redundancy of squaring. Consider the following equation:

$–5=5$ False. But square both sides, and we get...

25 = 25 True. So squaring both sides of a false equation can produce a true equation.

To see how this affects our equations, try plugging $x=\mathrm{-1}$ into the various steps of the first example.

## Why did we get a false answer of x=–1 in example 1?

 $\sqrt{x+2}+\mathrm{3x}=\mathrm{5x}+1$ Does $x=\mathrm{-1}$ work here? No, it does not. $\sqrt{x+2}=\mathrm{2x}+1$ How about here? No, $x=\mathrm{-1}$ produces the false equation 1=–1. $x+2={\left(\mathrm{2x}+1\right)}^{2}$ Suddenly, $x=\mathrm{-1}$ works. (Try it!)

When we squared both sides, we “lost” the difference between 1 and –1, and they “became equal.” From here on, when we solved, we ended up with $x=\mathrm{-1}$ as a valid solution.

Test your memory: When you square both sides of an equation, you can introduce false answers. We have encountered one other situation where good algebra can lead to a bad answer . When was it?

Answer: It was during the study of absolute value equations, such as $|\mathrm{2x}+3|=\mathrm{-11x}+42$ . In those equations, we also found the hard-and-fast rule that you must check your answers as the last step.

What do these two types of problem have in common? The function $|x|$ actually has a lot in common with ${x}^{2}$ . Both of them have the peculiar property that they always turn $\mathrm{-a}$ and $a$ into the same response. (For instance, if you plug –3 and 3 into the function, you get the same thing back.) This property is known as being an even function . Dealing with such “redundant” functions leads, in both cases, to the possibility of false answers.

The similarity between these two functions can also be seen in the graphs: although certainly not identical, they bear a striking resemblance to each other. In particular, both graphs are symmetric about the y-axis, which is the fingerprint of an “even function”.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!