We're going to continue solving equations and this time, we're going to solve some radical equations. We'll use our knowledge of radicals and powers to help us out. When we solve radical equations like this one, we want to isolate the square root or the root on one side of the equals sign. This equation already has the radical on one side of the equals sign. This radical is actually a grouping symbol, and the quantity p plus 1 is underneath of it. So, when we're trying to find the value of p, we want to undo the square root. To undo the square root, you want to square both sides of our equation. We know this since a square power undoes the square root. So, try finishing this problem. What would you get for p?
When we solve this radical equation, p should equal positive 8. Nice thinking if you got this one correct. When raising this left hand side to the 2nd power, we'll be left with the expression underneath the radical. So we'll just get p plus 1. On the right hand side, we'll square 3 to get a value of positive 9. We subtract 1 from both sides to isolate p, so 1p equals positive 8. One thing we want to do with radical equations is always check our answer. Sometimes this answer won't work in the original radical. Because sometimes it makes this radicand negative. We don't want that to be the case, so we can take the square root. So let's check for 8. We'll replace p with the value of 8, take the square root of 8 plus 1, which is 9, and that equals 3. We come out with a true statement on the end of 3 equals 3. So yes, this is the correct answer.
In general, we want to follow these four steps in order to solve radical equations. First, we want to isolate the radical on one side of the equation. Once that happens, we can raise both sides of the equation to the power that is equal to the index of the radical. Just like in the last problem. We wanted to undo a square root, so we used a square power. Then we can just solve the equation for the variable, and check our answer in the original equation. Keep in mind not to ignore this step. You really want to check your answers here, since sometimes they won't always work.
Whenever we solved radical equations, we want to isolate this radical first. So notice here that we have plus 5 next to this radical. We'll want to undo this by subtracting 5 from both sides in order to get this by itself. Once we subtract 5 from both sides, we'll have the cubed root of x plus 1 equal to negative 2. Now, you don't want to square this equation to solve it. What do you think we need to raise both sides by? We do need to raise it to a power, but it's not 2 in this case. Think about what you need to do and then solve for x. Be sure to check your answer and then, write your final answer over here. If there's no solution, then type, ns.
Here the correct answer is negative 9. Great work if you found this answer. If you were a little stumped, that's okay. Let's see how to solve it. We want to cube both sides of the equation, since a 3 is our index. A 3rd power will undo a cube root. So we'll have x plus 1 on the left, and negative 8 on the right. We need to make sure we cube this number as well, since we cubed this side. So negative 2 times negative 2 times negative 2 equals negative 8. Finally, we subtract 1 from both sides, to get 1x equals negative 9. Before we finalize this as our answer, we need to make sure we check it in the original equation. So I'll take negative 9 and substitute it in for x. This cube root only covers x plus 1. This means I only need to take the cube root of this expression, and then add 5. So we'll have the cube root of negative 8, then we'll add 5 and see if that equals 3. The cube root of negative 8 is negative 2, since negative 2 multiplied itself three times gives us negative 8. And finally, when we add here, our solution checks. In general when we solve, we want to keep in mind to isolate the radical first, then to raise both sides of the equation by the index of the root. And finally, once we have an answer, you want to check it to be sure we're correct.
Try solving this radial equation. When you think you know the value of x, enter it here. Or if you think there's no solution, type ns.
For this problem, x would equal positive 1. Nice effort if you got this one correct. To start solving this equation, we want to isolate the radical on one side, so we need to get rid of the subtraction of 1 here. We add 1 to both sides of the equation to get the 4th root of 3x minus 2 equals 1. We want to undo this 4th root. So we raise both sides of the equation to the 4th power. This leaves us with 3x minus 2 on the left, and 1 to the 4th, or just 1 on the right. We add 2 to both sides of the equation to get 3x equals positive 3. And finally, to get x by itself, we divide by the coefficient 3 to get x equals 1. To be sure this is correct, we check this in the original equation. We substitute x with the value of positive 1, and then try and check. Here, we'll get a 4th root, a positive 1. And we know the 4th root of 1 is just 1. Then we subtract to get 0 equals 0. So yes, this answer checks, which means this is our solution.
Now, let's make this a little bit more challenging. Sometimes, there might be two radicals in your equation. If there are two radicals, it's best to separate them, so one is on each side of this equals sign. We've seen adding numbers and fractions to both sides of an equation. Well, we can do the same with a radical. We'll just add it on the left, and add it on the right. On the left, we'll just have 2 root x. And on the right hand side, we'll have the square root of 3x plus 9. Now I definitely think you're at a position to solve this equation. Think about what you could do to both sides of the equation to eliminate the radical symbol. Then when you think you've solved for x, type your answer here. Again, if there's no solution, I want you to type ns. Good luck on this one.
The value of x that makes this original equation true is positive 9. Way to go if you got that one correct. It was tough. We can undo the square root on the left and undo the square root on the right by squaring both sides of our equation. And this is the step where you might have gotten stuck. We need to square this 2, and square this radical. 2 squared equals 4, and the square root of x squared is just x. This is why we'll have 4x on the left, and 3x plus 9 on the right. Next we can subtract 3x from both sides to get 1x is equal to positive 9. And finally, we want to check x equals 9 in the original equation to be sure we have the correct answer. Substituting 9 in here and here, will get this expression, which evaluates 0. Since 0 does equal 0, this solution checks and you can be sure that is our correct answer.
Here's our fifth problem on radical equations. What do you think the solution is for x? Again, if you're answer doesn't check, you want to type ns for no solution.
The only value of x that makes this equation true is x equals 3. Great work for getting that one correct. First we want to get each radical on one side of the equal sign. So we can add this radical to both sides of the equation. Now that each radical is on either side of the equal sign we can square both sides to undo these square roots. So on the left, we'll just be left with the radicand, Next, we just want to solve this linear equation for the value of x. So we subtract 2x from both sides and then, add positive 7 to both sides. This will leave us with 4x equals 12 and then, we divide both sides by 4 to get x equals equation. When we check x equals 3, we'll get this expression on the left, which evaluates to root 11 minus root 11, which equals 0. Our answer checks so we can use this as our solution set, x just equals positive 3.
What about this equation? What do you think the solution is here?
For this one, x would equal positive 21. Nice work if you got this one correct. Remember we always want to isolate the radical on one side of the equal sign. So we'll need to add 3 to both sides here in order to get the square root alone. So, with our square root on the left and positive 8 on the right, we're ready to square both sides to undo this square root. Squaring this side gives us 3x plus 1 and squaring this side gives us positive 64. Next we just solve for x. We subtract 1 from both sides and then divide the equation by 3. So 1x equals 21. We check our answer by plugging 21 into the original equation for X. When we check, we get the true statement 5 equals 5. So here is our solution.
How about this equation? What do you think x would equal in this case?
For this problem x equals positive 12. Great effort for getting this one correct. I hope you're on a roll with these. We start solving by isolating this cube root on one side of the equal sign. This means we'll have to subtract 5 from both sides, in order to isolate this cubed root. To undo this cube root, we raise both sides to the third power. The cube of the cube root x minus 4, is just x minus 4 and the cube of 2 equals 8. Now we're ready to solve for x and we just add 4 to both sides to get x equals 12 and as always, we want to make sure to check. We take the value of 12 and plug it in for x into the original equation. This cube root simplifies to the cube root of 8, which equals positive 2. 2 plus 5 equals 7. So yes, our solution checks.
We're almost done. Here's our eighth radical equation. What do you think the solution would be for a? Enter it here or type ns for no solution.
a equals 7 for this equation. Great work if you found this answer. This equation had two radicals. So, we want to split them up, so one radical is on each side of the equals sign. We add this square root to both sides of the equation to get our new one. Now we can undo these square roots by raising both sides to the second power. We'll have 4a minus 3 equal to 2a plus 11. We solved this linear equation by subtracting 2a from both sides, then adding 3 and dividing by 2. This gives us a is equal to 7. Now, like every problem before, you want to check this answer in the original equation. We plug in the value of which means our solution checks.
Here's our last problem for this lesson. What do you think the solution would be for this radical equation?
Here, a equals negative 3. Amazing work if you found this solution. This problem is a lot like the ones we've seen before. We want to move this radical to the other side of the equal sign. We do that by adding the square root of a plus 12 to both sides. Next we square both sides of the equation to undo these square roots. This gives us a linear equation of 2a plus 15 equals a plus 12. Next, we subtract a from both sides to get a plus 15 equals positive 12. Then we subtract 15 from both sides to isolate a, so a equals negative 3. Finally, we check our a value of negative 3 by replacing a in the original equation with that value. On the left-hand side of our equation, this simplifies to root 9 minus root 9. We know this is 0, or we could say 3 minus 3, which also equals