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Solving square-root equations: one solution

Sal solves the equation 3+√(5x+6)=12. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • mr pink red style avatar for user Gillian Bush
    What does he mean by "Principle square root"? Is it different from a regular square root?
    (101 votes)
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    • blobby green style avatar for user Balochfg
      If you square a positive number, like '2', you get '4' (positive). If you square the same number in negative form, like '-2', you also get a '4' (positive). So If you take the square root of a '4' you always get a '2' back. What you don't know is whether that '2' was originally a '-2' or a '(positive)2'.

      The "Principle square root" means you don't care about the sign, and you are only dealing in the positive domain. So Principle square root of '4' is just '2'.

      This becomes important when dealing with roots of variables. Because you literally don't know what the original number is and that is what you are solving for.

      Take for example the problem in this video. If he had not mentioned Principle Square root, then your X's answer could be either a negative number or a positive of that number. Plugging the negative or the positive numbers back in the original equation, you would get completely different results.

      So if he hadn't said "Principle Square Root", then X could have been either -15 or 15. If you plug in 15 back in the original equation it would check out. But -15 would get all sorts of crazy (try it). So in the end you would've known that the correct answer is 15. But he saved us the trouble of checking the original equation twice by saying "Principle Square Root" or, just the positive answer.
      (179 votes)
  • purple pi purple style avatar for user Victoria Fontaine
    What types of square roots are there??
    (6 votes)
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    • aqualine ultimate style avatar for user Selena Liu
      I think what Victoria means is that if there is a principle root, what kind of root is not a principle root? So let me explain. Let's assume x is positive. x^2=x*x , of course. But -x*-x=x^2. You can try this with any number. Let's say x^2=y. So the square root of y can be x or -x. The principle, or positive square root of y is x, but -x is the negative square root.(I'm not sure if you're supposed to say it like that, though. It's just how I say it.)
      I hope my explanation did more good than harm. :)
      (15 votes)
  • aqualine ultimate style avatar for user Niilo Niemi
    What does he mean about the negative square root and plus or minus square root at - ?
    (6 votes)
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    • old spice man green style avatar for user Petrie (Peter S. Asiain III)
      The plus or minus sign ± means that there are two possible square roots for a number, the positive and the negative square root
      For instance:
      ±√64=±8
      =+8 or -8
      Because (+8)^2=64 and (-8)^2=64
      Extra explanation:
      You might ask... Hey, how is that possible?
      Just think about this way, a positive multiplied by a positive will result to a positive and similarly a negative multiplied by a negative will result to a positive. In short if they have the same sign, they will result to positive number.
      (10 votes)
  • piceratops ultimate style avatar for user Erin
    What if a square root equals a negative number? Is it unsolvable? For example:
    √a(x)+b = -c
    a, b, c are definite numbers and x is the variable.
    Could you still solve the equation and isolate the variable?
    (5 votes)
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  • leaf red style avatar for user elliekat94
    How did he get 12 or x=15?
    (4 votes)
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    • purple pi purple style avatar for user Conny
      alright i'll solve that equation
      3+(5x+6)^(1/2)=12

      to solve it we need to get x alone on one side of the equal sign so to do that we'll subtract 3 from both sides of the equation so the 3 and -3 will cancel eachother on the left side of the equation
      (3+(5x+6)^(1/2))-3=12-3
      (5x+6)^(1/2)=9

      then we need to get rid of the square root over the x so to do that we'll raise both sides of the equation to the power of 2 because 2 is the inverse of 1/2 so when we raise (5x+6)^(1/2) the 2 and 1/2 will be multiplied and cancel eachother

      ((5x+6)^(1/2))^2=9^2
      (5x+6)^((1/2)*2)=9^2
      5x+6=9^2

      then we do like we did in the first step and subtract by 6 on both sides to move the 6 over to the right side and we'll get

      5x=(9^2)-6

      and finally we divide by 5 on both sides, then the fives in (5x)/5 will cancel eachother and we'll get
      x=((9^2)-6)/5 which is 15
      hope that helped
      (6 votes)
  • leafers ultimate style avatar for user Luke Butcher
    How can you tell if a radical equation, where the unknown is a radical, has no solution just by looking at it (or is this not possible)?
    Through experimentation I have found a way to do this but I'm not sure if it works 100% of the time. Is there no solution if: when the unknown is made the subject of the equation (but still under the square root), and the two sides of the equation have opposite signs?
    For example: sqrt(x) = -(5/7)
    Will all equations like this ^ have no solution?
    Hopefully this makes sense.
    (3 votes)
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    • aqualine ultimate style avatar for user Eric Tatum
      In general, when we solve radical equations, we often look for real solutions to the equations. So yes, you are correct that a radical equation with the square root of an unknown equal to a negative number will produce no solution. This also applies to radicals with other even indices, like 4th roots, 6th roots, etc. However, radicals with odd indices (like a cube root) can produce solutions when equal to a negative. For example, the cbrt(x) = -3 has the solution x = -27.
      (4 votes)
  • starky ultimate style avatar for user Howzieky
    Why can't you square both sides right off the bat (to get rid of the square root stuff), and then factor the polynomial that it gives you?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      Problem from the video: 3+√(5x+6)=12
      If you try to square both sides right way, the radical will not be eliminate. On the left side, you would have: [ 3+√(5x+6) ]^2
      To simplify this, you must use FOIL and it creates:
      9 + 3√(5x+6) + 3√(5x+6) + (5x+6) = 5x + 15 + 6√(5x+6)
      Notice, we still have a square root.

      The only way to make sure the square root is eliminated is to remove everything else from that side. So, Sal subtracted 3 prior to squaring the equation.

      Hope this helps.
      (5 votes)
  • mr pink red style avatar for user mflo
    what do i do after this : x squared (x2) =8
    (2 votes)
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    • female robot grace style avatar for user Zapzor
      If you mean x^2 = 8, then you have to transpose the square; look

      x^2 = 8
      x = sqrt(8)
      x = [sqrt(4)][sqrt(2)] // Using the product rule [ sqrt(a)*sqrt(b) = sqrt(ab)] look for perfect squares and simplify
      x = 2[sqrt(2)] // sqrt(4) = 2 since 2^2 =4
      and
      x = -2[sqrt(2)] // sqrt(4) = -2 since (-2)^2 is also equal to 4

      Therefore, the final and simplified answer is x = 2[sqrt(2)] and x = -2[sqrt(2)]

      *sqrt means square root
      (3 votes)
  • aqualine ultimate style avatar for user Haseeb
    When you put the square root symbol, what's the difference between the symbol for the +/- square root and the princible square root?
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      The principal square root is always the positive root. For example, the principal root of sqrt(9) = 3.
      But, sqrt(9) has 2 roots. If both roots are needed, usually when solving quadratic equations, then the +/- notation is used as a shorthand for saying we need +sqrt(9) = 3 and -sqrt(9) = -3
      (3 votes)
  • blobby green style avatar for user cconefrey-shinozaki
    In this video when Sal squares both sides, why doesn't he just multiply both sides by 5x+6?
    (3 votes)
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Video transcript

We're asked to solve the equation, 3 plus the principal square root of 5x plus 6 is equal to 12. And so the general strategy to solve this type of equation is to isolate the radical sign on one side of the equation and then you can square it to essentially get the radical sign to go away. But you have to be very careful there because when you square radical signs you actually lose the information that you were taking the principal square root. Not the negative square root or not the plus or minus square root. You are only taking the positive square root. And so when we get our final answer, we do have to check and make sure that it gels with taking the principal square root. So let's try. Let's see what I'm talking about. So the first thing I want to do is I want to isolate this on one side of the equation. And the best way to isolate that is to get rid of this 3. And the best way to get rid of the 3 is to subtract 3 from the left-hand side. And of course, if I do it on the left-hand side I also have to do it on the right-hand side. Otherwise, I would lose the ability to say that they're equal. And so the left-hand side right over here simplifies to the principal square root of 5x plus 6. And this is equal to 12 minus 3. This is equal to 9. And now, we can square both sides of this equation. So we could square the principal square root of 5x plus 6 and we can square 9. When you do this-- when you square this, you get 5x plus 6. If you square the square root of 5x plus 6, you're going to get 5x plus 6. And this is where we actually lost some information because we would have also gotten this if we squared the negative square root of 5x plus 6. And so that's why we have to be careful with the answers we get and actually make sure it works when the original equation was the principal square root. So we get 5x plus 6 on the left-hand side. And on the right-hand side we get 81. And now, this is just a straight up linear equation. We want to isolate the x terms. Let's subtract 6 from both sides. On the left-hand side, we have 5x and on the right-hand side, we have 75. And then we can divide both sides by 5. We get x is equal to-- let's see, it's 15, right? 5 times 10 is 50. 5 times 5 is 25 gives you 75. So we get x is equal to 15, but we need to make sure that this actually works for our original equation. Maybe this would have worked if this was the negative square root. So we need to make sure it actually works for the positive square root, for the principal square root. So let's apply it to our original equation. So we get 3 plus the principal square root of 5 times 15. So 75 plus 6. So I just took 5 times 15 over here. I put our solution in. It should be equal to 12. Or we get 3 plus square root of 75 plus 6 is 81 needs to be equal to 12. And this is the principal root of 81 so it's positive 9. So it's 3 plus 9 needs to be equal to 12, which is absolutely true. So we can feel pretty good about this answer.