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CCSS.Math:

we're asked to solve the equation three plus the principal square root of 5x plus 6 is equal to 12 and so the general strategy to solve this type of equation is to isolate the radical sign on one side of the equation and then you can square it to essentially get the radical sign to go away but you have to be very careful there because when you square radical signs you actually lose the information that you are taking the principal square root not the negative square root or not the plus or minus square root you're only taking the positive square root and so when we get our final answer we do have to check and make sure that it gels with taking the principal square root so let's try let's see what I'm talking about so the first thing I want to do is I want to isolate this on one side of the equation the best way to isolate that is to get rid of this 3 and the best way to get rid of the 3 is to subtract 3 from the left-hand side and of course if I do it on the left hand side I also have to do it on the right hand side otherwise I would lose the ability to say that they're equal and so the left hand side right over here simplifies to the principal square root of 5x plus 6 and this is equal to 12 minus 3 this is equal to 9 and now we can square both sides of this equation so we could square five the principal square root of 5x plus 6 and we can square nine when you do this when you do this when you square this you get 5x 5x plus 6 if you square the square root of 5x plus 6 you're going to get 5x plus 6 and this is where we actually lost some information because we would have also gotten this if we squared the negative square root of 5x plus 6 and so that's why we have to be careful with the answers we get and actually make sure it works when the original equation was the principal square root so we get 5x plus 6 on the left hand side and then on the right hand side we get 81 and now this is just a straight-up linear equation we want to isolate the X terms let's subtract 6 from both sides subtract 6 from both sides on the left hand side we have 5x and on the right hand side we have 75 and then we can divide both sides by 5 divide both sides by 5 we get X is equal to let's see X it fifteen right five times ten is fifty five times five is 25 this is 75 so we get X is equal to 15 but we want to we need to make sure that this actually works for our original equation maybe this would have maybe this would have worked if we were date if this was the negative square root so we need to make sure it actually works for the positive square root for the principal square root so let's apply it to our original equation so we get three plus the principal square root of five times 15 so 75 plus 675 plus six plus six so I just took five times 15 over here I put our solution in should be equal to 12 or we get three plus square root of 75 plus six is 81 it needs to be equal to 12 and this is the principal root of 81 so it's positive nine so it's three plus nine needs to be equal to 12 which is absolutely true so we can feel pretty good about this answer