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Radical and rational equations — Harder example

Video transcript

what is the sum of all the solutions to the above equation alright this is interesting we have three plus the square root of six m6m minus 26 is equal to M so anytime you have a radical equation like this it's a good idea to try to isolate the radical so I like to try to isolate the radical first so let's try to subtract 3 from both sides to just be left with just the radical on the left hand side and so 3 minus 3 is 0 and on the left hand side you're just left with the square root of 6 M minus 26 and that's going to be equal to M minus 3 now to get rid of the radical we can square both sides of this equation so we'll square that side and then we can square that side as well and so the square root of 6 M minus 26 squared that's going to be 6 M minus 26 and then M minus 3 squared that's going to be M squared minus 6 M it's going to be minus 3 M minus 3 M plus 9 and if this step you found a little confusing I encourage you on Khan Academy to review multiplying expressions or I'll do a quick primer right here M minus 3 times M minus 3 that's the same thing as M minus 3 squared you'll have M times M which is M Squared M times negative 3 which is negative 3 M negative 3 times M which is negative 3 M and then negative 3 times negative 3 which is positive 9 and the negative 3 M plus a negative 3 n that is negative 6 M and I encourage you especially on cos on the SCT we'd have to do some time pressure to be able to do things like square a binomial like this very quickly to realize it was gonna be M Squared 2 plus 2 times the product of these the product of these is negative 3m 2 times that is negative 6 M and the negative 3 times negative 3 is positive 9 so now let's see if we can solve for M so let's get all of let's get all of our everything on the right-hand side of the equation so let's subtract 6 M from both sides so I'm going to subtract 6 7 from both sides and let's add six to both sides and the whole reason why I did that is to just clean out what I have on the left-hand side so on the left-hand side I'm just gonna be left with zero and on the right-hand side I'm gonna have M squared minus twelve M plus 35 or I could write M squared minus twelve M plus 35 is equal to zero and so let's think about the M's that satisfy this well I can factor this if I think about well what two numbers have a product of 35 but if I were to sum them I get to negative 12 well 35 I could think of seven times five but seven plus five is positive 12 but what about negative 7 times negative five well their products going to be positive 35 and negative 7 plus negative five is negative 12 so I could factor this into M minus five times M minus seven is equal to zero and if this step does not make sense to you where I factored this quadratic encourage you to look up factoring quadratics on Khan Academy to get a little more practice doing that but if I have the product of two things that equal zero that means that one or both of them need to be equal to zero so M minus 5 could be 0 or M minus 7 would be equal to 0 well to make M minus 5 equal to 0 you just add 5 to both sides you would have M is equal to 5 or same thing over here add 7 to both sides M is equal to 7 so the two the two solutions are 5 & 7 and if you want the sum of all the solutions it's going to be 5 plus 7 is equal to 12 and if you just want to verify that these actually work try them out right over here 6 times 5 is going to be 30 minus 26 which is 4 the principal root of 4 is positive 2 3 plus 2 is equal to 5 it's equal to M and if M was 7 if M was 7 this is going to be 3 plus the square root of 42 minus 26 is 16 and this needs to be equal to 7 well the principal root of 16 is four and three plus four is indeed equal to seven