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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.

## Want to join the conversation?

• so plus or minus is basically like the math equivalent to
Schrödinger's cat?
• You could say that. It's also like a quantum computer where it's binary can be a 0 or a 1 but it doesn't chose until you oberve it
• "Quad" is the start of a couple of different latin words - one that means four and a few others that relate to squares. In the case of quadratics, the quad- bit relates to the fact that the equations contain squares (and often relate to square shapes too).

Hope that helps - there's a more in-depth explanation here: http://mathforum.org/library/drmath/view/52572.html
• Sal does the simple ones, but he doesn't show us how to do these: h(x)=−5x^2+180
• It is the same principle, set h(x)=0. This gives 0=-5x^2+180, then add 5x^2 to both sides to get 5x^2=180, divide by 5 to get x^2=36, so x = ±√36. This gives two solutions, x=-6 and x=6.
• 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?
(x-3)/8=2/(x-3)
• You can also cross multiply, which means you would get (x-3)(x-3)=16, the you would get x^2-6x+9=16, then x^2-6x-7, then (x-7)(x+1) so that the solutions would be +7 and -1
• 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?
• 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.
• I am struggling to solve these problems I struggle with the 2nd step
• The first two steps are exactly like solving a two step linear equation. Two step equations require you to add/subtract first and then divide or multiply second. If you have x/2+5=13, you subtract 5 on both sides to get x/2=8, then opposite of divide is multiply by 2 to get x=16. Or if you have 3x - 2 = 10, add 2 to get 3x=12, divide by 3 to get x=4.
The only difference in the video is the third step of taking the square root, so x^2/2 + 5 = 13 gives x^2=16 giving x=+/- 4.
• Can somebody give me the definition of a Quadratic
(1 vote)
• A quadratic is a 2nd degree polynomial, basically any polynomial that can be written in the form: Ax^2+Bx+C is a polynomial. A, B, and C are real numbers, and A can't = 0 (because if it did, you wouldn't have a 2nd degree polynomial).
• how do you know when to use plus or minus for what X equals. Do you just choose either?
• 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!
• Is there a place I can do some practice problems for each lesson?
• Its really great because you can watch the video if your stuck or even get hints

## 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.