 This video will talk about solving radical equations. So if you remember when we were solving quadratic equations, we had an x squared and we took the square root of both sides and the square and the square root cancelled each other out and gave us an x. Well now we're going to start out with a square root, and we want to get rid of the square root sign so it should make sense that we would want to then square both sides, and in this case just square the square root. Again the square and the square root cancelled each other out and we're just going to have x plus 5. And down here again the square and the square root cancelled each other out and we're just left with what's underneath, 2y minus 1. So how does that apply with an equation? Well remember what you do to one side you have to do to the other side, so if I want to get rid of the square root by squaring on this side I have to square on this side as well. The square cancels the square root leaving us with x and 4 squared is 16. It's not plus or minus this time because we know it has to be a positive underneath the radical, so it's just x equals 16. If we get a more involved thing underneath the radical we still just square both sides. So the square cancels the square root leaving us with x minus 6 and then 2 squared would be 4. We have to solve for x so we have to add the 6 to both sides and x is going to be equal to 10. Now we always have to check these answers. So the square root of 16 is it equal to 4? Yes, so we know that's our answer. And over here the square root of 10 minus 6 is that equal to 2? Well the square root of 10 minus 6 would be the square root of 4 and we all know that the square root of 4 is 2. So now we are certain that x equals 10. So let's try again. Square both sides. Square cancels the square root and we have 2A plus 1 equal to 5 squared or 25. Subtract 1 from both sides. 2A is going to be equal to 24. Divide by 2. A is going to be equal to 12. And then we check. Remember you have to check. It's not an option here. So the square root of 2 times RA which is 12 plus 1 should be equal to 5 if we did it right. Well this gives us the square root of 24 plus 1 or the square root of 25 and we know that the square root of 25 is equal to 5. So A does equal 12. Over here we can square both sides. The square cancels the square root leaving us with x and negative 9 times negative 9 is going to be 81. And then we look at it and say is the square root of, this is our check, is the square root of 81 equal to negative 9? No. What is the square root of 81? It's positive 9. And positive 9 does not equal negative 9. So here we have no solution. And now you can see why you have to check. Alright, let's do steps to solve this equation. We have an extra step here. First we have to isolate the radical. So we have to subtract 2 from both sides. So we have the square root of x plus 7 equal to 8. Then you want to square both sides. So we square the square root and we square the 8. And then you just want to solve. So the square cancels the square root leaving us with x plus 7 and 8 squared is 64. And if we subtract 7 from both sides then x, these are going to cancel each other out, is equal to 57. And we check. Square root of 57 plus 7. And then if we add 2 to that we should get 10. This is the square root of 64 plus 2 is equal to 10. And the square root of 64 is 8. And we all know that 8 plus 2 is equal to 10. Alright, so try again. We have to isolate the radical. So we want to subtract 8 from both sides. So we have the square root of 6y plus 1 equal to negative 7. Then we want to square both sides. So the square cancels the square root and we have 6y plus 1 equal to 49. And then we want to solve. So we subtract 1 from both sides. 6y is going to be equal to 48. And if we divide by 6, y is going to be equal to 8. So we come over here to check it. And we say the square root of 6 times 8 plus 1. And if we add 8 to it it should be equal to 1. If we did it right and there's a solution. This is the square root of 48 plus 1 plus 8 equal to 1. This is the square root of 49. And the square root of 49 is 7. And 7 plus 8 does not equal 1. So this can't be a solution. There is no solution. Finally, we need to add 12 to both sides. Remember isolate the radical. So we have the square root of 4a is equal to adding 12 would be positive 6. And then we're going to square both sides. Square cancels the square root. And we have 4a. 6 squared is 36. And then we solve. All we have to do is divide by 4 to solve. And a is going to be equal to 9. And then we check. The square root of 4 times 9. And then if we subtract 12 from that do we get a negative 6? Well, square root of 4 times 9 is going to be the square root of 36 minus 12. Equal to negative 6. And 36 square root of 36 is 6 minus 12. Is a negative 6. So we know that a equal 9.