in this lesson we're going to focus on solving radical equations so let's start with this example the square root of 3x plus 1 is equal to 4. if you know what to do feel free to pause the video and work on this example so what's the first thing that we need to do the first thing is we need to get rid of the square root and the only way to do that is to square both sides so on the left side we now have is 3x plus one on the right side four squared four times four is sixteen and now let's subtract both sides by one so what we now have is three x is equal to sixteen minus one which is fifteen and then divide both sides by three so x is equal to five and then if you want to check your work so using the original equation let's replace x with five so we have three times five plus one three times five is fifteen fifteen plus one is sixteen and the square root of sixteen is four so five is indeed an answer so x equals five number two square root seven minus x plus three is equal to five so what's the first thing we should do in this example now we don't want to square both sides yet we want to move the 3 from the left side to the right side so let's begin by subtracting both sides by 3. so five minus three is equal to two now at this point you wanna take the square of both sides if you did it before you would have to foil root seven minus x plus three but now you don't have to do that so on the left is just gonna be seven minus x and two squared is four so now let's subtract both sides by seven and so negative x is four minus seven or negative 3 and if we divide both sides by a negative 1 negative 3 divided by negative 1 is positive 3 and so that's the answer and you can check if you plug it in 7 minus 3 is 4 the square root of 4 is 2 2 plus 3 is 5. so that's going to work try this one the cube root of x plus 15 let's say it's equal to 3. so this time instead of squaring both sides we want to raise both sides to the third power so that the index number of three will cancel with the exponent three and so it's going to be x plus 15 is equal to 3 to the third power which is 27 and now all we need to do is subtract both sides by 15 and so x is 27 minus 15 which is 12 and so that's the answer for that one and what about this one 2 square root x is equal to x what is the value of x in this example we can square both sides the 2 is multiplied to root x and so we don't have to foil anything 2 squared is 4 and the square of a square root will cancel the square root so it's just going to be x on the right side we have x squared now let's subtract 4x from both sides so on the left they will cancel we're going to have 0 on the left on the right x squared minus 4x and which we can factor that we can take out the gcf which is x and so we can see that x is equal to zero and x is equal to four now let's check both answers let's go back to the original equation so when x is zero the square root of zero is simply zero so zero equals zero the equation is true now what about when x is four the square root of 4 is 2 and 2 times 2 is 4. so this works as well therefore both answers are correct here's another one x raised to the one fourth plus four is equal to seven so let's begin by subtracting both sides by four so x to the one fourth will be equal to seven minus four which is three so now to get rid of the one-fourth exponent we need to raise both sides to the fourth power one-fourth times four is just one and three to the fourth that's gonna be 81 that's three times three times three times three you can see it this way three times three is nine and the other two threes there are nine nine times nine is eighty one now what about this one the square root of 3x plus 4 let's say it's equal to the square root of 4x plus 3. go ahead and find the value of x so in this example we're going to take the square of both sides and so the radicals will disappear therefore three x plus four will be equal to four x plus three so now let's subtract both sides by three x and let's subtract both sides by three four minus three is one and four x minus three x is one x or simply x so x is equal to one try this one what would you do in this example if you have two radicals and a number you don't want to square both sides with one side containing both radicals you want to move one of the radicals to the other side and let's say this is negative two by the way let's move this radical to this side so this is going to be root four x plus nine and then minus two it's going to be positive on the right side now what we can do is take the square both sides on the left side we only have a radical and once we square it it's simply going to be 5 plus x on the right side because we're dealing with a subtraction side that separates the radical and the 2 we have to foil it's going to be root four x plus nine minus two times another root four x plus nine minus two so once we multiply these two radical four x plus nine times radical four x plus nine the radical will cancel and we're just gonna get four x plus nine the radical times negative two that's going to be negative 2 root 4 x plus 9 and we're going to get another similar term once we multiply these two together and then finally negative 2 times negative 2 is positive 4. now let's go ahead and combine like terms so we can combine 9 and 4 which is thirteen so we have four x plus thirteen and we can also combine these two which is going to be negative four root four x plus nine now because we have another radical we need to get this radical by itself on one side of the equation so what i'm going to do i'm going to subtract both sides by x and by 5. and then i'm going to take this term and move it to the left side so it's going to become positive so on the left we have positive 4 root 4 x plus 9 and on the right 4x minus x is 3x 13 minus five is positive eight so now let's go ahead and square both sides so we can completely get rid of the radical the square of four is sixteen we don't have to foil this term because the four is multiplied to the radical the square of root four x plus nine is just going to be four x plus nine the square will cancel the radical now three x plus eight squared we need to foil it due to the plus sign that separates the 3x and the 8. so let's write it like this for now on the left let's distribute 16. 16 times 4 that's 64. and sixteen times nine that's a hundred and forty-four now let's foil three x times three x is nine x squared 3x times 8 that's 24x 8 times 3x is another 24x and finally 8 times 8 is 64. so now let's combine like terms so we have 64x plus 144 and that's equal to 9x squared plus 48x plus 64. so let's get rid of this and let's subtract both sides by 64x and also by 144 so on the left it's gonna be zero and then we have 9x squared 48 minus 64 is 16. so negative 16x and 64 minus 144 that's negative 80. so can we factor this particular trinomial what would you say 9 times negative eighty is negative seven hundred and twenty what two numbers multiply to negative seven twenty and add to negative sixteen let's see factors of nine are three and three factors of eighty ten and eight factors of ten five and two factors of eight four and two and so forth this can help us to find factors of seven twenty so if we use 10 this would be 10 and negative 72 which doesn't differ by negative 16 so we have to increase it so 15 goes into it 15 is 5 times 3 if we divide 720 by 15 that's going to be negative 48 so we're getting closer but we're not quite there yet so let's see we also have uh 8 times 3 that's 24 we could try that negative 720 divided by 24 that's negative 30. that differs by six so we went too far we need something between 15 and 24. so let's see what numbers can we choose between 15 and 24 well we could try 18 that's 3 times 3 times 2. that's eighteen so negative seven twenty divided by eighteen that's uh negative forty so those two differ by twenty two so we need something between 18 and 24 we could try 10 times 2 which is 20. actually let's see if that's going to work 720 divided by 20 that's 36 so that works that's what we need so let's replace 16 x squared rather negative 16x squared with negative 36x and positive 20x and then let's factor by grouping so now let's get rid of this stuff so in the first two terms let's take out the gcf which is going to be 9x by the way this problem is too difficult to factor you can also use the quadratic formula if we take out 9x it's going to be x minus negative 36x divided by 9 or divided by 9x that's going to be negative 4x and from the last two terms the greatest common factor is 20. 20x divided by 20 is x negative 80 divided by 20 is negative 4 and this should be just negative 4. that x shouldn't be there let's get rid of that negative 36 x divided by 9x is simply negative 4. in the first parentheses we're going to have x minus 4 and inside the second one the stuff on the outside is going to go into it that's 9x plus 20. so to solve we need to set each factor equal to zero so in the first one we can see that x is equal to four and the second one we can move the 20 to the other side it's going to be negative 20 and then divide by 9. so these are the two answers but now let's make sure both answers are indeed correct so x is equal to 4 and negative 20 over 9. and the original problem is root 5 plus x plus 2 and that's equal to root four x plus nine so let's replace x with four five plus four that's nine four times four is sixteen plus nine is twenty-five the square root of nine is three and the square root of twenty-five is five three plus two is five so the equation is balanced which means that x is indeed equal to four now let's try negative 20 over 9. i'm going to use the calculator for this so 5 minus 20 over 9 that's 25 over nine if you take the square root of that this will give you five over three and four times negative twenty over nine that's negative eighty over nine plus nine that's one over nine and the square root of one over nine is one over three and you could tell that this is not going to be equal to each other five over three plus two that's 11 over three which is about 3.67 that doesn't equal 0.33 so therefore the second answer is an extraneous solution it doesn't work and so this is why you need to check your answers especially when you have multiple answers and if you're solving radical equations sometimes one of the answer might be an extraneous solution