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Current time:0:00Total duration:4:57

Equation that has a specific extraneous solution

CCSS.Math:

Video transcript

we're asked which value for D we see D in this equation here makes x equals negative 3 an extraneous solution for this radical equation square root of 3x plus 25 is equal to D plus 2x and I encourage you to pause the video and try to think about it on your own before we work through it together all right now let's work through this together so the first thing that just to remind ourselves is what is an extraneous solution well that's a solution that we get or we think we get but it's really just a byproduct of how we solved it but it isn't going to be an actual solution of our original equation now how do these extraneous solutions pop up well it pops up when you when you take the square of both sides so for this equation right over here to get rid of the radical I'd want to square both sides of it if I square both sides the left-hand side will become 3x plus 25 and the right-hand side if I square this is going to be what it's going to be d squared plus 4 for DX plus x squared so that's just squaring both sides of this but notice you actually there's actually a different equation than this one that if you squared both sides you would also get this what is that different equation well the different equation is if you took the negative of one of these sides so for example if you had if you start at the original equation the negative square root of 3x plus 25 plus 25 is equal to D plus 2x you square both sides of this you still get this purple equation because you square a negative you get a positive so both of them when you square both sides get us get us over here and so when you solve this purple equation this is going this is a quadratic right over here you just rearranged it a little bit you get into standard quadratic form you'll get two solutions and it turns out one of the solutions is going to be for this yellow equation and one of the solutions is going to be for the purple equation and if if the X the one the solution that is for the purple equation is going to be an extraneous solution for the yellow equation it's actually not going to be a solution for the yellow equation so when they say which value for D makes x equals negative 3 and extraneous solution for this yellow equation that's the same thing as saying what value of D makes x equals negative 3 a solution for this so a solution a solution for this if it's a solution for this it's going to be an extraneous solution for that because these are two different equations we have this is we're taking the negative of just one side of this equation to get this one if you took the negative of both sides of this and that becomes the same thing you could multiply both sides of an equation times a negative value so a solution for this which is equivalently a solution which is equivalent to a solution if I instead of putting the negative on the left-hand side if I multiply the right-hand side by the negative but anyway let's think about which value for D makes x equals negative 3 a solution for this well let's substitute x equals negative 3 here and then we just have to solve for D if x equals negative 3 is going to be negative the square root of 3 times negative 3 is negative 9 plus 25 is equal to d 2 times negative 3 is negative 6 so d minus 6 and so now we can square both sides of we can square actually let's do it this way we can I don't want to square both sides because we lose some information there's going to be the negative square root of negative 9 plus 25 is 16 is equal to D minus 6 so this is going to be equal to the negative 4 principal root of 16 is 4 we have the negative out front is equal to D minus 6 and then add 6 to both sides you get 2 is equal to 2 is equal to D so if D is equal to 2 here if D is equal to 2 then a solution to this purple equation is going to be x equals negative 3 and so that would be an extraneous solution because if x equals negative 3 satisfies this over here it's definitely going to satisfy this over here but it's not going to satisfy this up here and you could verify this if this is equal to if this is equal to 2 try out x equals negative 3 you're going to get on the left-hand side are going to get 16 and on the right-hand side you're going to get 2 minus 6 which is equal to negative 4 2 minus 6 which is negative 4 so this does not work out x equals negative 3 is not a solution to this but it is a solution for this and it is a solution to this quadratic right over here so D equals 2 makes X equal negative 3 an extraneous solution for this equation