welcome to today's lesson it's 5.4 solving radical equations so we want to get rid of the radical today here are three easy steps that we can see to solve our equations so for example or in this example example one here we want to isolate the radical so you want to make sure that this piece gets on the side by itself so you take care of all other operations first so here i'm going to divide by 2 so the twos cancel i get the square root of x plus one equals two then to undo a square root this is the new piece you square each side if this was a cube or if this was to the fifth root you would take it to the fifth exponent so then you raise it to that exponent and these cancel you get x plus one equals two squared is 4 then continue on solving your equation so x here equals 3. now i always like to check my answers so i'm going to go 2 square roots of 3 plus one and ask myself does that equal four so i have three um plus one which is four so i have 2 times the square root of 4. well 2 times the square root of 4 well that's 2 and that equals 4 so it checks i did my work correctly now if i move to the one on the right hand side here i am going to have to isolate oops wait a second my pen is not working um it just jumped on me i apologize [Music] all right we can continue so here i want to isolate my cube root and what's under there so then i take care of my 1 and i add the 1 add the 1. so i get the cube root of 2x minus 9 uh equal to 3 but now to get rid of the cube i have to cube both sides so that these cancel off so i get 2 x minus 9 and i'm cubing that 3 on the other side which gives me 27 i add 9 i add 9 i get my 36 is equal to 2x divide by 2 divide by 2 and x equals 18. if i go ahead and i substitute that in here cube root of 2 times the quantity 18 minus 9 and then minus the 1. does that equal 2 well that's our question so i take the cube root of 36 minus 9. 36 minus nine that's the cube root of 27 and then you take away one we're still asking ourselves if it equals two so cube root of 27 three minus one yes that certainly checks it is two so let's go to our next slide we get into a little more difficult but actually it ends up being simpler than you think now we have two radicals ones on each side so they're isolated but since one is isolated on one side and the other one's on the other side you basically just take care of the square root so you jump to the hey i'm gonna square both sides and if you square both sides you'll notice the square root cancels on the left side but the same is true on the right hand side so then we go ahead and solve this like a linear equation and then subtract 5 subtract 5 x is equal to 2. again i'm gonna go check this make sure it works because sometimes your answers don't work when they are radicals because you have a square root of a negative number so i'm going to go over here 2 times 2 plus five and we're gonna ask ourselves is that equal to square root of two plus seven so two or two times two is four four plus the five is nine so we have square root of 9 which is 3. over here we have the square root of 9 when you add 2 and 7 so we know there they're equal but we can also take the square root so it checks so now let's move on where we don't have the actual square root written in there but we have it as an exponent a rational exponent so technically speaking in this problem this is my radical so i subtract two yet i subtract two i get two x raised to the three fourths equal to eight but then if i go ahead and take a look at this i need to get rid of that fraction so to get rid of the fraction i take the fraction to the reciprocal take to the reciprocal if i take it to the reciprocal which would be four thirds in this example what's gonna happen is these two exponents cancel but i have to take it to the four thirds over here now to be quite honest i go ahead and i say go ahead and use your calculator so 2x is equal to and when you put that in your calculator just make sure you go ahead and you put parentheses around your fraction so this is equal to sixteen i go ahead divide by two divided by two x is equal to eight so if you put that in your calculator or in into check you have two times eight but that is all raised to the 3 4 and then you're adding 2 to that and your question is does it equal 10 so if you put this in your calculator as 16 raised to the 3 4 well if you take the fourth root of 16 that's two two to the third is eight so eight plus two does equal ten it checks now some people you know might not understand that right away but right here we have another example but now we have some parentheses with it so if you take this whole thing to the reciprocal the reciprocal of one-half one-half is two over one so i'm going to take this to the two over one or simply 2 and then these cancel so you have x plus 30 equals x squared well now if you have a quadratic this is where you have to bring everything over to set it equal to zero so if i do this i have zero is equal to x squared minus x minus thirty well now if you take the factors of thirty negative thirty that is 1 30 5 6 you want one so because this will give you 29 so you want to use the 5 and the 6 so i get 0 is equal to x minus 6 and x plus 5 because when i add them together i need this to be a positive this to be a negative to get the negative 1. now if i go ahead and solve this i'm going to have 6 oops x equals six x equals negative five now what i would do is i would i would put those in there and see which ones work so if you do this and put it back into your equations like you're checking it you're going to have 6 plus oops it's not registering give me a moment six plus thirty and that to the one half and the one half is like saying the square root so that's like thirty six to the one half well that's going to give you six if you come over here that's supposed to equal the x because that's what we have on the right hand side our question was do they equal and they certainly do okay so then you try your negative 5 so then i have negative 5 plus 30 to the one-half let's check if that equals negative five well if you put these two in there you're going to get uh 30 take away five or 25 and if you take that to the one half you get five and we want it to equal negative 5. so since it does not equal negative 5 they're not equal this value is extraneous so that concludes our lesson that's probably the toughest it gets that concludes our lesson for today so therefore we'll start our homework