Solve each equation in Exercises 92–93 by making an appropriate s...

Question

Solve each equation in Exercises 92–93 by making an appropriate substitution. x^4 - 5x^2 + 4 = 0

Accepted Answer

Welcome back. I'm so glad you're here. We are solving for X. And we get to use substitution. So let's check out our problem. We've got four X. To the fourth minus 37 X. Squared plus nine equals zero. Okay and when we're using substitution we want to look at this middle term here X squared and say can we take the middle term and can we square it to get our first term? And yep X squared squared does get us X. To the fourth. So what we can do now is substitute something else for this X. Squared and you can pick whatever you want. I'm gonna pick W. And so instead we're going to have a W. And we're going to square it to get the first term. So instead of four X to the fourth we've got four W squared minus instead of 37 X squared. It's 37 W. Plus nine equals zero. And great. That's something that we can factor. So how do we factor this one? This one's it's a little bit different because there's that number in front of the W. Squared. So we could have four W. Times one. W. Those could be negative. We could have two W. Times two. W. Those could be negative but a clue we've got here is that we've got a four And in nine and they have to multiply and get to add up to be negative. 37 four times nine is already. 36. So that's pretty close to 37. Let's play with this a little bit to see if the four. And the nine can work out. So we'll do the four W. Times one W. And then to get the nine we'll keep that nine in the position where it will be multiplied by the four. And they have to be positive when multiplied and negative when added. So let's make them both negative. And sorry that was nine times one will get us that nine. And let's see if this works. We'll do a quick foil to check. So first four W times W will before W squared the outsides four W. Times negative nine will be negative 36 W. The insides negative one times W. Is negative one. W. And the last negative one times negative nine is plus nine, combine our middle terms four W squared minus W. Plus nine, yep. That checks out. So we've got two factors that we can set equal to zero. We've got four W minus one. We can set equal to zero and we've got W minus nine. We can set equal to zero and we can get rid of the parentheses. And we've got that. We want to get the W. By itself. We can add wonderful sides. So we've got four W equals one and divide both sides by four and one of our Ws is 1/ over here. We'll add nine to both sides. So we get W equals nine. Cool, we've got both of our Ws but we're not solving for W. We're solving for X. So we've got to plug both of these in up here at this equation. So we've got X squared equals well one of our ws is nine and we've got X squared equals the other W. Is 1/4. How do we get that? X by itself? We'll take the square root of both sides, so squared of X squared is X. And then when we take the square root of a number, we have to do plus or minus and the square root of nine is three. There we got two, answers three and negative three. And now we've got to take the square root of both sides of this one. And now we've got X equals square root of one is one square to four is two. And we have to remember that plus minus. So we've got negative three negative one half, one half and three. We look at our answer choices that matches with answer choice. D. Well done. We'll catch you on the next one.

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