Solve each equation. See Example 2. 3x^2/(x-1) + 2 = x/(x-1)

Question

Accepted Answer

Hey everyone in this problem for the following equation were asked to solve for x and were given five X squared over X minus five plus six is equal to X. Over x minus five. So writing out what were given five, X squared over x minus five plus six is equal to X over x minus five. Now, in order to solve, Okay we want to combine this into one function or one equation. Okay, one rational expression. And in order to do that we need a common denominator. Okay, this is just like adding numbers. Okay we want a common denominator, so how can we do that? Well we have x minus five in the denominator in this left term we have x minus five in the denominator on this far right term. Okay so let's get a denominator of x minus five for our plus six term. We're gonna have five X squared over x minus five and then we're gonna multiply six by x minus five. Okay. Over X -5. Okay, so we're multiplying by X -5, divided by X -5 which is just one. So we're not changing the equation. Alright and then we get equals two x minus sorry X over x minus five. Now we have a common denominator across all three of our terms, we can move them all to one side, we can combine them together. So moving to the left hand side, we're gonna have five X squared expanding this term, we get six X ok minus six times negative or sorry plus six times negative five which is gonna give us minus and then subtracting this term from the right hand side minus X. And all of this is over a common denominator of x minus five and it's all going to be equal to zero? Yes. Alright so let's just simplify that numerator five X squared plus five X minus 30 Over X -5 is equal to zero. Now this is starting to look more like an equation that we're comfortable solving. Okay we have this quadratic in the numerator um equal to zero. Okay so that's starting to look familiar. So let's notice that we have this common factor of five in the numerator we have five X squared five X and minus 30 which is divisible by five. So let's go ahead and divide this entire equation, the left side and the right side by five. If we divide the left hand side by five we get X squared plus x minus six Over x minus five is equal to and then we have zero divided by five which is just gonna be zero. Alright so we have X squared plus x minus six divided by x minus five is equal to zero. Okay how can we have this equation equal to zero? Well if the numerator is equal to zero then this entire thing is going to be equal to zero. So if we have x squared plus x minus six equal to zero then this is going to satisfy our equation. However we have to be careful. Okay in our original equation we're dividing by x minus five. Okay so we know that we cannot have x equal to five. If we have x equal to five the denominator will be zero and the function or the expression will not exist. Okay so we want to find when X solves X squared plus x minus six equals zero. But we want the value of X not to be equal to five. Okay. So it's important to put that restriction so that when we get our answers we can check to make sure that they are Okay. Alright so now we just have this quadratic equal to zero. This is just like finding the roots of an equation. Okay so we can go ahead and do that. Let's factor this. So we're gonna have X plus three times X minus two is equal to zero. And we know that we get a result from each factor okay if one factor is equal to zero then the entire thing is equal to zero because they're being multiplied. So we have X equals from the first factor negative three. Okay, X plus three equals zero means that X is equal to negative three and from the second factor we have X minus two equal to zero, so X is equal to two. So the two solutions that we find our X is equal to negative three and two and you'll notice that neither of them are equal to five, so they're both okay, they're not gonna cause any issues. And so we have our solution X. Is equal to negative three or positive too. If we go up to our answer choices, we see that that is going to be answer choice. D. Thanks everyone for watching. I hope this video helped see you in the next one.

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Solve each equation. See Example 2. 3x^2/(x-1) + 2 = x/(x-1)