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Sal solves the equation 3+√(5x+6)=12. Created by Sal Khan and Monterey Institute for Technology and Education.
Video transcript
We're asked to solve the equation "three plus the principle square root of five x plus six is equal to twelve" 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 were taking the principle 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 it gels with taking the principle 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 three. And the best way to get rid of the three is to subtract three 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 are equal. And so the left-hand side right over here simplifies to the principle square root of five x plus six. And this is equal to Twelve minus three. This is equal to nine. And now we can square both sides of this equation. So we can square five, the principle square root of five x plus six and we can square nine. When you do this, when you do this, when you square this, you get five x, five x plus six. If you square the square root of five x plus six you're going to get five x plus six! And this is where we actually lost some information, because we would have also gotten this if we squared the negative square root of five x plus six. 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 principle square root. So we get five x plus six on the left-hand side, and on the right-hand side we get eighty-one. And now this is just a straight-up linear equation. We want to isolate the x terms. We'll subtract six from both sides. Subtract six from both sides. On the left-hand side we have five x, and on the right-hand side we have seventy-five. And then we can divide both sides by five. Divide both sides by five, we get x is equal to... Let's see.. x it-it's fifteen. Right? Five times ten is fifty, five times five is twenty-five, which is seventy-five. So we get x is equal to fifteen. But we want - 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 taking - If this was the negative square root. So we need to make sure it actually works for the positive square root, for the principle square root. So let's apply it to our original equation. So we get three plus the principle square root of five times fifteen. So seventy-five plus six, seventy-five plus six plus six, so I just took five times fifteen over here, I've put our solution in, should be equal to twelve. So we get three plus square root of seventy-five plus six is eighty-one it's equal to twelve. And this is the principle root of eighty-one, so it's positive nine. So it's three plus nine needs to be equal to twelve, which is absolutely true. So we can feel pretty good about this answer.