Solve each equation in Exercises 41–60 by making an appropriate s...

Question

Solve each equation in Exercises 41–60 by making an appropriate substitution.

9x^4 = 25x^2 - 16

Accepted Answer

Welcome back. I'm so glad you're here. We've got this equation. We have four X to the fourth equals 61 X squared minus 225. And we are to solve for this equation using the substitution method. Well, for the substitution method, we first want to get it looking as close to a quadratic equation as we can. So I want to bring over the 61 X squared in the negative 225. So that the right hand side is going to be equal to zero. And this will keep our four X to the fourth, which is the one. The variable with the highest degree will keep that positive. So it will be four X to the fourth minus 61 X squared plus 225 equals zero. Great. Now the substitution, what we do for substitution is when this is really close to a factory ble quadratic equation. We look at the first and the second variables and we see if the first variables exponent. This four is double that of the second variables exponents the two. And yes it is. So then we take the second variable X squared. And we set it to any variable that we like. I mean not itself, but in this case we'll use W. So we're going to use a w for the X squared. And how about the X to the fourth? Will we know if we took X squared and squared it, what we do to one side. We have to do the other x squared squared is X to the fourth, which is exactly what we're looking to replace and that will be replaced with W squared. So now we can bring down the rest four W squared minus 61 W plus 225 equals zero. Great. This is something we can factor and yet, I mean it looks kind of nasty. There are a few steps to factoring this one but we can do it okay for our first so we have to have something times something equaling for W squared. And we've got options, we can have a four W times a W or we could have a two W times a two W and those would both get us for W squared. How about over here? This positive 225. Well to get 225 we could have the factors of four times or excuse me five times, 45. We could do nine times 25, we could do three times 75 or we could do 15 times 15 because if you broke down 225 in a factor tree, it's going to be three times three times five times five. And those are all the various combinations of those factors in two factors. So now we've got to take the factors of 225 And find out how we can combine those two give us this negative 61. When we go through the foiling method So negative 61, we notice that the numbers have to add up to be a negative number. So where we can influence that is by making the factors that multiply to equal 225. Both negative because then they'll still be a positive 225 but they'll add up to a -61. So negative 61 that's a pretty big number to get down to negative 61. We don't want to overshoot it because both numbers will be negative. And so that rules out this negative three times negative 75 negative 75 is already lower than negative 61. So adding anything else is just going to be more negative when they're both negative numbers. So now we just have to deal with these three and we can either choose to start with the fours and the ones or the twos and the twos and you could just take negative five and negative 45 go ahead and foil those out and see if they work. But let's be a little strategic about this here instead of guess and check and well we want to get pretty deep with the four Times something but we don't want to take a four times one of these big numbers that's going to overshoot 61 again. So let's take a four times these first factors which would give us a one times these other factors. So four times negative five is negative 20 and one times negative 45 is negative 45 nope, those two do not add up to negative 61. Let's try the next row. Four times negative nine is negative 1 times negative 25 is negative 25. And yeah, hey look, those do add up to negative 61. So we are dealing with negative nine and negative as are other factors for our lasts and four W. And one W as are factors for our first. So let's be sure we pair these up correctly. The four W has to be multiplied by the negative nine. So we'll put four W here and we'll put negative nine here. So that those two get multiplied. Oops, I wanted to do green four W. Times a negative nine. When we do our Outsides and then we just fill in their partners. So four W is going to be times W. And the negative nine is multiplied by the negative 25 to get 225. And if that was a bit of a trip and you need to foil to double check, we can do that real quick. Our first four W. Times W. Is four W squared. Our Outsides four W. Times negative nine is negative 36 W. Insides negative 25 times W is negative 25 W. And r lasts negative 25 times negative nine is positive, 225 combining like terms in the middle, we have four W squared minus 61 W plus 225. Yes, that matches our equation from before. So that means we can proceed yes, there are still even more steps to take here. I'm going to take out this section of work and now we're going to take each of our factors that we discerned which are four, W minus 25 W minus nine. And we're going to set both of those equal to zero so that we can solve for w the left hand side. Four, W minus 25 will add 25 to both sides. So we've got four. W Equals 25. Now let's divide both sides by four. And we have W equals 4th. There's one of our ws Well, W minus nine. That's a little bit easier to solve a add nine to both sides. So instead of W minus nine equals zero, it's W equals nine. Well, yeah, except we're not trying to solve for W. We're trying to solve for X. So we have to go back to the beginning and substitute an X squared anywhere. We have a W. So not W equals 25 4th, it's X squared equals 4th and instead of W equals nine it's X squared equals nine. How do we get that? X by itself? While we're going to take the square root of X squared. And what we do to one side, we have to do the other. So no longer X or this no longer X squared. But now the squared of X squared is just X equals. And we have the square root of 25 over the square root of four. Well the square root of 25 is five and the square to four is to but because we took the square root we have to have plus or minus. So we've got a positive five halves and a negative five halves so far. Now again with the X squared equals nine, we're going to take the square root of both sides of the equation and square root of X squared is just X. The square root of nine is going to be plus or minus three. So we also have an option of negative three and three. We have four options for X. We look at our answer choices and that matches answer choice. A well done. We'll catch you on the next one.

Solve each equation in Exercises 41–60 by making an appropriate substitution. 9x^4 = 25x^2 - 16

Key Concepts

Polynomial Equations

Substitution Method

Quadratic Formula

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