A Nice square root math question|United States|Math olympiad|can you solve.

Hi everyone, let's solve for the value of a. First of all, let us square both sides of this equation. So if we do that, we're going to have this side squared and this side squared, right? Now remember that when you have a plus b squared, that is equal to a squared plus 2ab plus b squared. And in this case, our a is the square root of a, and our b is the square root of a minus 8, right? Alright, so we're going to substitute this into the formula. That means we can write this as squared. That's the square root of a squared plus 2ab plus 2 multiplied by the square root of a times the square root of a minus 8, right? Then plus b squared; that should be the square root of a minus 8 squared equals 4 squared, which is 16. Now, if we solve this, we are going to have, after canceling this, a plus. This will give us 2 because they share the same radical sign, which is the square root. It's going to be the square root of a multiplied by a minus 8, right? Then, plus, this will eliminate the square root sign; we have minus 8 equals 16. Let's collect like terms. We are going to have a plus this a minus this 8 plus 2 root a bracket a minus 8 is equal to 16. So we have a, sorry, this is a, yes, a plus this a. So a plus a will give us 2a. Then minus 8 plus 2 times the square root of a times a is a squared, and a times minus 8 is minus 8a, which is equal to 16. Now let's set this aside, the one with the square root, and move all the others to the other side of the equation. If we do that, we are going to have 2 root a squared minus 8a equal to 16. Now 2a will cross over to become minus 2a, and minus 8 will cross over to become plus 8. So we are going to have 2 root a squared minus 8a equals 16 plus 8, which gives us 24 minus 2a. Now that we have 2 here and this side can also be divided by 2, let's divide 2 by 2. Divided by 2, divided by 2, divided by 2, right? So this is going to cancel this, and we have the square root of a squared minus 8a equal to 12 minus what? a. So, to eliminate this square root symbol, let us square both sides of this equation. So if we square both sides of the equation, we will have the square of the square root of a squared minus 8a equal to 12 minus a squared. Now we are going to, this is going to remove this. Now we are going to have a squared minus 8a equal to, then this place will give us a squared minus 2ab plus b squared. So it's going to be 12 squared minus 2 times 12 times a plus b squared, which is plus a squared. So we have a squared minus 8a equal to 144 minus 24a plus a squared. Now let's combine like terms. Let's combine like terms. So here we have a squared minus 8a. Let's transfer this to become minus 144, transfer this to become plus 24a, transfer this to become minus a squared is equal to 0. Now at this point, a squared minus a squared will give us 0. Then minus a plus 24a is the same as 24a minus 8a, which will give us 16a, right? So we have 16a minus 144 equals 0. Now 16a equals 144. So if we divide this side by 16 and this side by 16, we get that a is equal to 9. Now this is the value of a. Now let's check if this is correct. So let's verify. Let us verify. If this is your first time visiting this channel, please click the subscription button and turn on the notification bell so you don't miss our videos. We upload videos every day, so please tune in at your convenience to watch our latest uploads, and give us a thumbs up if you're genuinely enjoying this video. So we have the square root of a plus the square root of a minus 8 equals 4 squared. That is the question. Is it equal to 4? Sorry. Now, the square root of a means the square root of 9 plus the square root of 9 minus 8 equals 4. So the square root of 9 is 3, plus 9 minus 8 is 1, which equals 4. So we have that 3 plus 1, because the square root of 1 is still 1, is equal to 4. So we have that 4 is equal to 4. Now you can see that the left side is equal to the right side, which confirms that the problem or the solution is correct. Thank you so much for watching. Goodbye.