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We're asked to solve the equation

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"three plus the principle square root of five x plus six is equal to twelve"

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And so the general strategy to solve this type of

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equation, is to isolate the radical sign on one side

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of the equation, and then you can square it to

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essentially get the radical sign to go away.

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But you have to be very careful there, because when you

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square radical signs you actually lose the information that

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you were taking the principle square root, not the

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negative square root or not the plus-or-minus square root.

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You're only taking the positive square root. And so

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when we get our final answer, we do have to check

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and make sure it gels with taking the principle square root.

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So let's try, let's see what I'm talking about.

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So the first thing I want to do, is I want to isolate

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this on one side of the equation. The best way to isolate that

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is to get rid of this three. And the best way to get rid of the

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three is to subtract three from the left hand side. And of

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course if I do it on the left-hand side I also have to do

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it on the right-hand side. Otherwise I would

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lose the ability to say that they are equal. And so the

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left-hand side right over here simplifies to the

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principle square root of five x plus six. And this is equal to

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Twelve minus three. This is equal to nine. And now we can

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square both sides of this equation. So we can square

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five, the principle square root of five x plus six

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and we can square nine. When you do this, when you do this,

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when you square this, you get five x, five x plus six.

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If you square the square root of five x plus six you're

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going to get five x plus six! And this is where we actually

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lost some information, because we would have also gotten

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this if we squared the negative square root of five x plus six.

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And so that's why we have to be careful with the answers

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we get, and actually make sure it works when the original

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equation was the principle square root. So we get

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five x plus six on the left-hand side, and on the right-hand side

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we get eighty-one. And now this is just a straight-up linear

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equation. We want to isolate the x terms. We'll subtract

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six from both sides. Subtract six from both sides.

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On the left-hand side we have five x, and on the right-hand

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side we have seventy-five. And then we can divide both sides by five.

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Divide both sides by five, we get x is equal to... Let's see..

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x it-it's fifteen. Right? Five times ten is fifty,

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five times five is twenty-five, which is seventy-five.

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So we get x is equal to fifteen. But we want - We need to

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make sure that this actually works for our original equation.

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Maybe this would have, maybe this would have

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worked if we were taking - If this was the negative

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square root. So we need to make sure it actually works

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for the positive square root, for the principle square root.

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So let's apply it to our original equation. So we get

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three plus the principle square root of five times fifteen.

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So seventy-five plus six, seventy-five plus six

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plus six, so I just took five times fifteen over here, I've put

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our solution in, should be equal to twelve. So we get

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three plus square root of seventy-five plus six is eighty-one

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it's equal to twelve. And this is the principle root of

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eighty-one, so it's positive nine. So it's three plus nine

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needs to be equal to twelve, which is absolutely true.

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So we can feel pretty good about this answer.