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”.

what is variations in raman spectra for nanomaterials
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
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
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
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 did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!