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Int Algebra: Solving Equations with Two Radicals


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About this Lesson

  • Type: Video Tutorial
  • Length: 11:22
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 122 MB
  • Posted: 12/02/2008

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra: Equations & Inequalities (50 lessons, $65.34)
Intermediate Algebra Review (25 lessons, $49.50)
College Algebra: Radical Equations (4 lessons, $5.94)

In this lesson, Professor Burger will show you how to solve equations that contain two radicals (roots). When you have an equation with two square roots, you'll want to have them on opposite sides of the equal sign. Then, you'll square both sides of the equation. If there is still a radical remaining, you'll have to isolate it on one side of the equation and then square both sides once again. There will be several examples included in this lesson that will show you how to approach this type of problem and then how to check your work.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Intermediate Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers real numbers, equations and inequalities, exponents and polynomials, rational expressions, roots and radicals, relations and functions, the straight line, systems of equations, quadratic equations and quadratic inequalities, conic sections, inverse and exponential and logarithmic functions, and a variety of other AP algebra and advanced algebra.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

2174 lessons

Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit or visit Thinkwell's Video Lesson Store at

Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...


Recent Reviews

~ lmcgrath

I am feeling more confident about my exam now. I am taking College Algebra online and am teaching myself from the book. It works out just fine until my textbook decides to skip steps an assumes that I know how we got to the next step!

This video saved my life! THank you!!!!

I get it!
~ birniesj

Thank you so much, You make it so easy to understand...I appreciate it :)

~ lmcgrath

I am feeling more confident about my exam now. I am taking College Algebra online and am teaching myself from the book. It works out just fine until my textbook decides to skip steps an assumes that I know how we got to the next step!

This video saved my life! THank you!!!!

I get it!
~ birniesj

Thank you so much, You make it so easy to understand...I appreciate it :)

Roots and Radicals
Equations with Radicals
Solving Equations with Two Radical Expressions Page [1 of 2]
Now I thought I would show you a real, really, really radical, radical equation. So radical, radical. You've heard of
pizza pizza? This is radical radical. And the reason why it’s radical radical is because, in fact, not surprisingly, there
are two radicals. So here’s the equation that I want us to now solve. It’s the square root of 2x+3=1+ x+1.
Two square roots, one equation. How do you handle this? Well, if you remember what I was talking about earlier,
you may recall that, in fact, to get rid of a square root we have to square both sides. But also remember that I said
that if, in fact, you square something like this, you've got to foil and by foiling then you still have more square roots in
the problem. So this seems like a catch-22 because if I bring this one over to the other side, then when I foil this side
I'm going to be trouble. And if I bring this square root over so I have the two square roots together, when I square I'll
have still have a foiling dilemma. So this looks like I'm all foiled up. So what would I do? Well, the answer is you're
going to have to square this thing out twice. The first time you square should get rid of one of radicals and then once
you get rid of one of the radicals, on the other side you’ll have to foil. You’ll get a whole bunch of messy stuff and
they’re will be one square root in there. Then you do the same again. Bring everything else over to the other side and
then square out again. So this is going to be a two squared method.
All right. So let’s now square both sides of this. If I square this side here, that square is going to kill the square root,
but the thing is I'm also going to square this side and when I square this side, I'm going to have to foil that thing out.
So, in fact, if I square this side I see the following; 2x + 3. However, now I've got to square this thing out. Now that’s
going to actually require a little teeny bit of work. So let me actually do that out here for you because this might be
involved. This may get ugly. Let’s think about this. So what I want to do is square out or multiply 1 + x +1 by
itself. Now there is such a temptation to make the number one classic mistake that I just want to mention it once
again. The number 1 classic mistake, there it is, the squaring mistake. Saying that this just equals 12 plus this thing
squared. Wrong. There’s those middle terms we have to foil. So always remember foil. So let’s foil out. 1 x 1 is 1
so this equals 1. Now the middle term notice is just 1 times this so it is the x +1 . But notice the outside term is
also x +1 so I have two of them. So I see +2 x +1 and then what’s the last times the last? Well, that’s going to
be the
2 x +1 . The
2 x +1 . And what happens? Well, the square root and the square, they kill each other. They
cancel each other out so it lifts the radical and so I'm left with this equals 1 + 2 x +1 plus and then I just have an x
+1. So, in fact, I can do a little bit of combining here. The 1 and the 1, that makes 2. Let me write the x over here,
too. And then I'm left with the 2 x +1. So, in fact, when you square off the right hand side of this, what you end up
with is 2+x+2 x+1. So that’s the answer. So that’s what I'm going to write in here right now. So I see 2—
whoops! Plus, that’s a plus sign, not a little star, x+2 x+1. I just carefully squared that out and got this except for
that little font mistake. Is everyone happy?
So I'm going to just get rid of all this other work and notice what happens. Well, it looks really, really complicated,
especially with that funny looking plus sign, but the important thing to notice is just count how many square roots you
have. The original problem I had 2. Now I only have 1. So now I'm going to repeat this awful process again. So
what am I going to do? I'm going to take all the terms that don’t have the square root with it like these guys and I'm
going to bring them over to here and I'm going to square root again. So let’s do that right now. If I bring this x over to
the other side, that’s a plus so I have to subtract it. So I have a 2x minus this x would just give me an x. If I bring this
2 over to the other side, I'd subtract 2 from both sides. If I bring over the -2, I have a 3 minus that 2. That would be a
+1. So that would equal 2 x +1. Now you may be wondering the following. Gee, what should I do with that 2? It’s
not in the square root. So is that going to be a problem? Well, if you don’t want it there, you can divide both sides by
2 and have it here and in the denominator, but if you think about it, if I have any multiple in front of a square root,
when I square it, I'm not going to have to foil. I only foil when I'm adding something. So, in fact, the squaring—this is
going to be completely fine. This will produce a 4 when I square it and then the radical would lift here. So either way
is fine. I don’t like dealing with fractions so I'm just going to take this and square it as it is. I'm not adding anything
else to this so I should be fine. So once again, I square both sides. See? With two radicals I square twice. Guess
how many times you have to square with three radicals? I'm not even going to tell you. Here we go. If I square this
Roots and Radicals
Equations with Radicals
Solving Equations with Two Radical Expressions Page [2 of 2]
out, I'm going to see x2 + 2x + 1. Look how quickly I square that out. Are you impressed? I hope you're impressed
because it took me years to go through that.
Now when I square this out we have to be a little careful. I have a product of two things. Remember how the laws of
exponents go. I have to take this thing and square it and then take this thing and square it and multiply them together.
So I have 22 which is 4 and then I'm going to have the
x +12 and so what happens? That just lifts the radical. So I
have this. But now, as is, that is wrong. And do you see why? I'm making the classic mistake. It’s usually with a
negative sign out front here. It’s my classic mistake #4. I call it the subtracting mistake or just the distributing mistake.
The point is you have to remember that that 4 has to hit everything. You have to spread that 4 and make sure it hits
everything. So you have to put parenthesis right there. This is so important. I hope you make this mistake with me
right now so you’ll never make it again. All right. That 22 has to hit everything in the square root.
Well, on with that, I can solve this. This thing here if I distribute that I'm going to see what? Let me just write this out
here maybe. This is the same thing as 4x + 4, just distributing. Well, now I want to solve this. So what do I do? I
see it’s quadratic. I pull everything over to the left-hand side, have it equal 0, try to factor, try to solve, this is here.
Got the idea? So what do I have here? I have x2. If I take this 4x, remember that’s just this part here so you can
ignore that now if you want. If I take this 4x and subtract it over, I see 2x - 4 x gives me a negative 2x and then here I
have a +4 which when I bring over becomes a -4 but I still have a +1. So that produces a net gain of -3 and that
equals 0. So here’s the quadratic that I now have to solve. Remember, that quadratic came from looking at this
original really complicated equation. So can I factor this? Well, I hope I can. Let’s try it and see. I see x2 so I'll put an
x and x. This negative sign tells me signs will be opposite since there’s no coefficients in front of the x. I could put
them anywhere I want, plus or minus let’s say. Two numbers that multiply to give 3, but then subtract to give 2. It
looks like 3 and 1 are a good choice. Should I put the 3 here? Well, no, because that’s the bigger term and when I
would subtract I would get a +2. So I'll put the 3 here, put the 1 here. Notice that’s a -3x + x is a minus -2x. Perfect.
So for these two things to multiply to give 0, either this factor is 0 or that factor is 0. If this factor is 0, can you see that
means that x must be equal to -1 because x + 1 = 0 means that x must be -1. If x - 3 = 0, if I bring the 3 over, I see
that x has to equal 3. So there’s two solutions to this quadratic, the question is, are there two solutions to the original
question? Remember the original question, by the way, didn’t have those squares there. In fact, you know what?
Now I'm sort of regretting that I wrote those squares right on top there. In fact, let me just take those squares off just
because--remember the original equation did not have those squares. But with the magic of web technology, folks,
using high tech technology, in fact, I can just edit those things out and you won’t see that. See the power of the web?
The information superhighway working for you right now.
Let’s try and see if any of these answers are correct. So I'm going to plug in a -1 for x on the left and on the right. If I
plug it in on the left, what do I see? I see the square root of, well, -1 x 2 is -2 and then I add a +3 and so what does
that equal? Well that equals the square root of 1, which equals 1. Now does that equal—this is the question,
1+ -1+1. Well -1 + 1 is just 0. So, in fact, this just equals 1 + 0, which equals 1 and they are equal. So this does
check. So, in fact, this is an answer. It checks.
So now you're saying, fine. If that checks, this one is not going to check. So let’s just save time and say it’s not an
answer and move on, right? Well, wrong. We need to check to make sure. Oops! Oh, my god. Don’t look at the 2.
That was a little witch in the web. You know a web? No, it’s me. Here we go. Let’s try now putting in a 3. If I put in
a 3 here, I'll see the 2 3 x is 6 so this is 6 + 3 and that equals the 9 and the 9 I know is 3. So on the left-hand side
I have 3. Now what about on the right hand side? On the right hand side I want to know if this is the same. I have
1+ 3+1. Well, 3 + 1 I know is the 4 . The 4 is 2. So I see 1 + 2--look at that! This equals 3. These things
actually check as well. So, in fact, this is another answer. So this particular rational expression has two answers; x =
-1 and x = 3. So the moral of this story is always remember if you have a lot of radicals to slowly peel them off by
separating them as slowly as you can, isolating the square root and then once you get an answer, always remember
check it back to the original one, not this one, bleep! You see, I can put it back if I want. But instead, check it to the
original one and make sure they’re still O.K. All right. See you next.

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