 This video will deal with the square root property. So we have our x's here. We want to fill in our table because this is going to lead us into solving these equations. So 2 squared is going to be 4 and then the square root of x squared is really undoing. They undo each other. So if I undo 4, I go back to the 2 and 3 squared would be 9. This is really saying that the square root of 4 over here was equal to 2. 3 squared equal 9, so the square root of 9 would take us back to the 3. 6 squared is 36 and really saying the square root of 36, well 6 times itself is 36, so we get back to 6 and 10 squared would be 100. So the square undoes the square root, so we're just back to 10 and then 11 would be 121 and the square root of 121 is going to be 11. Another way to think about this is the square root of 4 is the square root of 2 squared and the square root of 2 squared, the square root, cancel each other out and you have 2. Again this would be like the square root of 3 squared, but the square and the square root cancel each other out and you're just left with 3. So the square root property states that if we have an x squared equal to k, or k is some constant, then when we solve we can take the square root of both sides because the square root of x squared will give us just plain old x and that's what we're trying to solve for. But we have to remember, do not forget this plus or minus. We have to have both cases because we don't know if a positive is squared to get a positive number or a negative is squared to get a positive number. So we have the square root, or x squared equals 64, so it's a constant on this side equal to x squared over here. So we can take the square root of both sides and this will give me x because the square and the square root cancel each other out and that's plus or minus and the square root of 64, 8 square root of 64 so it would be plus or minus 8. Now we have a problem because I have both the constant and the x squared on one side so I need to take my constant to the other side so I have to add the 36 to both sides. So positive 36 goes on this side and now I'm ready to take the square root of both sides so the square root of x squared equal to plus or minus the square root of 36. Well the square and the square root cancel each other out so we have x equal to plus or minus and we know what the square root of 36 is, that happens to be 6. Now we have our x squared term and our constant on the opposite sides but I've got this coefficient that I don't like. I didn't say a x squared it said x squared equal k so I have to divide both sides by 3 before I can do anything else. So I have to get the square variable all by itself and then I'm ready to say the square root of x squared is equal to plus or minus the square root of 9 and again the square and the square root cancel each other out leaving me with x equal to plus or minus and square root of 9 would be 3, 3 times 3 is 9. Now this looks like a cute little graphic but what I'm trying to show you is it doesn't matter what squared on this side if it's equal to k if I take the square root of both sides plus or minus on the constant side then I'm just going to have this nice little cute happy face because the square is going to cancel the square root equal to plus or minus k I can only draw as cute as that thing is. So when we have something that looks like this it's something squared all by itself equal to 7 or a constant so I can take the square root of the something as long as it doesn't have a coefficient on it anything squared and then plus or minus the square root of 7 and the square is going to cancel the square root leaving us with the x minus 9 and then we have plus or minus 7 square root of 7. Well now I have to get x by itself but if I'm subtracting 9 remember that just means that I have to add 9 to both sides so this gives me x and on this side the square root of 7 and 9 I can't combine them because they're not both integers so I have 9 plus or minus and I don't know what the square root of 7 is so I just leave it as the square root of 7. So x is equal to 9 plus or minus the square root of 7. Again I have a quantity squared there's nothing outside of the parentheses so this too is a coefficient is okay. I'm just ready to take the square root of both sides now. So the square root of 2x minus 3 quantity squared is equal to plus or minus the square root of 16 my pins are starting to drag. Square cancels the square root so we have what's underneath the radical equal to plus or minus and I know the square root of 16 so right away I'm going to say plus or minus 4. Now it's just an equation I'm really solving 2x minus 3 plus or equal to positive 4 and 2x minus 3 equal to negative 4 so add 3 to both sides 2x will be equal to 7 and when I divide by 2 x is equal to 7 halves that's one of my answers and I add 3 to both sides again that this time I've got negative 4 on the other side so 2x will be equal to negative 1 divided by 2 and x is equal to negative 1 half so what happens when I have x squared equal to a negative number if I take the square root of both sides square root of x squared is equal to plus or minus the square root of negative 7 let me rewrite that 7 negative 7 then this square root will cancel the square again and leave me with x but the square root of negative 7 is undefined it's not a real number okay we could also say it's not real so if that's the case there is no solution I can't answer it with a real number so I don't have a solution so anytime you end up with x squared equal to a negative number you can't have an answer because you can't take the square root of a negative number now over here it looks like our difference of squares because they're both perfect squares but this little sum is going to make all the difference if we go back and think about it a would be x x times x would be x squared and b would be 5 but I have a positive 5 which means that this has to be the same sign as the middle and that's kind of funny anyway I'd have x and x and so a lot of people would want to say plus 5 and plus 5 since it's a positive 25 but if I do my outside terms that gives me 5x and if I do my inside terms that's another 5x and it should be 10x which is not what my problem has so you cannot factor this would be prime or no solution actually would probably be better because you cannot factor x squared plus a perfect square