In Exercises 91–100, find all values of x satisfying the given co...

Question

In Exercises 91–100, find all values of x satisfying the given conditions.

y = x^3 + 4x^2 - x + 6 and y = 10

Accepted Answer

Welcome back. I'm so glad you're here. We are given this equation, we have Y equals X cubed minus five X squared minus 121 X plus 700. We are to solve for x. We're all given this information that Y equals 95. So we can go ahead and substitute that in for y. Now we've got 95 equals X cubed minus five X squared minus 121 X plus 700. And I want to bring that 95 over to the right hand side of the equation, setting the left hand side equal to zero. Because perhaps then we can do some factoring. So we now have zero equals X cubed minus five X squared minus 121 X plus 700 minus 95 is 605. Great. Well we've got four terms. So we can give it a go with some factoring by grouping group the first two terms together and then the second two terms together. And see if there's anything we can pull out. Looking at each set of two terms for the first set, X cubed minus five X squared. Those will have an X squared. So we can factor that out an X cubed divided by x squared. Leaves us with X negative five X cubed divided by X. Excuse me. Negative five X squared divided by X squared. Leaves us with negative five. That first set is done. Now looking at the second set. And well, let's figure out what we've got here. We know that 121 creating a factor tree. Well that's 11 times 11. So we could try since 11 is the only factor of 100 and 21. Um we could try dividing 605 by 11. And in fact, yeah, it is divisible by 11, it's 11 times 55. And we could even go further and say that 55 is times five. So 605 is 11 squared times five And 121 is 11 squared. Actually 605 is divisible by 121. So we can factor out a negative 121 here and negative 121 X divided by negative 121 that just leaves us with X and 605 divided by negative 121 gives us a negative five and look at that, these are the same X minus five. So we can factor out an x minus five because both our x squared and r negative 121 are being multiplied by x minus five. Great. Now we're not quite done because we look at this X squared minus 121 and we notice that that is a difference of squares that is x squared minus squared. And we recall from previous lessons that when we have a difference of squares, when we have a squared minus B squared, that equals A plus B times A minus B. So in this case real quick just to reiterate that A is X and B is 11. Now we can go back over here and we can factor x squared minus 121. So the A plus B. That's going to be X plus 11 and a minus B is going to be x minus 11. And don't forget to bring down our friend x minus five. Well this is great. We've got three factors and now we can set each one of these equal to zero. X minus five equals zero. X plus 11 equals zero. An X -11 Equals zero. And we can add five to both sides for x minus five equals zero. Which will give us X equals five. There's one of our values for X plus 11. We can subtract 11 from both sides which will give us X equals negative 11 for one of our values. And for this final one we can add 11 to both sides. For x minus 11 equals zero And that will give us x equals 11. We look at our answer choices and this matches with answer choice. C. Well done. We'll catch you on the next one

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