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Solving quadratics by taking square roots

Sal solves the equation 2x^2+3=75 by isolating x^2 and taking the square root of both sides. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user Benjamin Nation
    so plus or minus is basically like the math equivalent to
    Schrödinger's cat?
    (26 votes)
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  • male robot johnny style avatar for user Afif Al Mamun
    Why quadratic means 2, whether the word quad means 4? I'm confused about this, anyone can explain please?
    (0 votes)
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  • duskpin ultimate style avatar for user Lorien Lorien
    Where would you use Quadratics in real life? Like I know that algebra is really important, but are Quadratics specifically really going to be necessary when I'm not in school anymore?
    (5 votes)
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  • blobby green style avatar for user sharon
    Can't seem to have any idea about this one
    (x-3)/8=2/(x-3)
    (3 votes)
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  • starky sapling style avatar for user Hodorious
    So, we have x^2 = 36

    Could've we just take the principal square root of both sides, and then end up with |x| = 6 → x = ±6?
    (2 votes)
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    • aqualine ultimate style avatar for user Shenana
      Yes absolutely.
      That is actually what happens every time we take the square root of both sides.
      Because mathematicians are lazy, we don't want to solve the absolute value equation, so we skip that step and jump straight to x = _+ 6, because that is what we will get.
      In other words, what you did is 100% correct, but by jumping straight to x = _+ 6, you can skip a step.
      (3 votes)
  • blobby green style avatar for user jones63dawn
    Is there a place I can do some practice problems for each lesson?
    (2 votes)
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  • scuttlebug blue style avatar for user sude06
    I have a question. So to isolate x when it comes to x^2, we use inverse operation and square root it. So when we try to isolate x when it comes x^3, what do we do? Do we, instead of square rooting it, cube root it?
    (3 votes)
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  • primosaur ultimate style avatar for user NEOVISION
    doesn't square roots only take positive numbers?
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      If you are working only with real numbers, then we can't take the square root of a negative. We say there is "no real number solution". We don't say there is no solution because a solution does exist if you use complex numbers.
      (2 votes)
  • mr pink green style avatar for user Gauri Patel
    could someone help me? i am only in 6th grad and i don't exactly understand this. is it just like solving a regular equation or is there more to it?
    (2 votes)
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    • female robot amelia style avatar for user E.
      If by "regular equation" you mean, for example, 4x-8=32, it is definitely like solving a regular equation. It just uses squares and square roots, so if you simplified an equation down to x^2 = 9, you could just take the square root of both sides (this "un-squares" it :D) and you would get x = 3.

      Does that help?

      --
      WriterScientist
      (2 votes)
  • mr pants teal style avatar for user s23gnovello
    how do you know when to use plus or minus for what X equals. Do you just choose either?
    (1 vote)
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    • hopper cool style avatar for user Iron Programming
      When taking square roots the answer can either be a positive or negative number, so we use the plus or minus symbol to represent that.

      Which value we use actually depends on the problem we are doing. If say, we are calculating a value for something like 'years since 2000' then that value can only be positive.

      So if we are only dealing with positive values, we say we are taking the principle square root, which means we only use the positive value.

      To be completely precise in math, we need to account for the different and include +/- before the number to make sure everyone looking at our value knows the actual value(s) we are dealing with.

      Now, like I said if the value can't be negative then there is no reason to include the negative sign there.

      Hope this helps!
      (3 votes)

Video transcript

We're asked to solve the equation 2x squared plus 3 is equal to 75. So in this situation, it looks like we might be able to isolate the x squared pretty simply. Because there's only one term that involves an x here. It's only this x squared term. So let's try to do that. So let me just rewrite it. We have 2x squared plus 3 is equal to 75. And we're going to try to isolate this x squared over here. And the best way to do that, or at least the first step, would be to subtract 3 from both sides of this equation. So let's subtract 3 from both sides. The left hand side, we're just left with 2x squared. That was the whole point of subtracting 3 from both sides. And on the right hand side, 75 minus 3 is 72. Now, I want to isolate this x squared. I have a 2x squared here. So I could have just an x squared here if I divide this side or really both sides by 2. Anything I do to one side, I have to do to the other side if I want to maintain the equality. So the left side, just becomes x squared. And the right hand side is 72 divided by 2 is 36. So we're left with x squared is equal to 36. And then to solve for x, we can take the positive, the plus or minus square root of both sides. So we could say the plus or-- let me write it this way-- If we take the square root of both sides, we would get x is equal to the plus or minus square root of 36, which is equal to plus or minus 6. Let me just write that on another line. So x is equal to plus or minus 6. And remember here, if something squared is equal to 36, that something could be the negative version or the positive version. It could be the principal root or it could be the negative root. Both negative 6 squared is 36 and positive 6 squared is 36, so both of these work. And you could put them back into the original equation to verify it. Let's do that. If you say 2 times 6 squared plus 3, that's 2 times 36, which is 72 plus 3 is 75. So that works. If you put negative 6 in there, you're going to get the exact same result. Because negative 6 squared is also 36. 2 times 36 is 72 plus 3 is 75.