Use the graph of y=2/x^2 - x below to solve the equation 4x^3-10x^2+2=0.

[Kulla_9289]

@Dr.Peterson How would you solve [imath]4x^3-10x^2+2=0[/imath] by using the graph of [imath]y=2/x^2 - x[/imath] by using the method presented by the book?

 

[Dr.Peterson]

@Dr.Peterson How would you solve [imath]4x^3-10x^2+2=0[/imath] by using the graph of [imath]y=2/x^2 - x[/imath] by using the method presented by the book?

This is the problem we started with, right?

I'd start with [imath]4x^3-10x^2+2=0[/imath], see that it is a cubic and that the highest degree in [imath]y=2/x^2 - x[/imath] is 1; so I'd want to make the former look more like the latter by dividing the entire equation by ...

I'll let you decide that. I insist on seeing some work on your part before I fully answer any more questions. You are being remarkably resistant to my efforts to get you to do anything.

 

[Kulla_9289]

How is the degree 1?

So, 4x^5 - 10x^4 + 2x^2 = 0 and y = 2 - x?
Then, 4x^5 - 10x^4 + 2x^2 - x = - x and y = 2

 

[Dr.Peterson]

How is the degree 1?

I said,

... the highest degree in [imath]y=2/x^2 - x[/imath] is 1

How is that not obvious?

The first term is 2x^-2, with degree -2 (to the extent that the term "degree" applies); the second is -x, with degree 1.

So, 4x^5 - 10x^4 + 2x^2 = 0 and y = 2 - x?
Then, 4x^5 - 10x^4 + 2x^2 - x = - x and y = 2

Why in the world would you do that? Perhaps you don't know the meaning of "divide".

I said,

I'd start with [imath]4x^3-10x^2+2=0[/imath], see that it is a cubic and that the highest degree in [imath]y=2/x^2 - x[/imath] is 1; so I'd want to make the former look more like the latter by dividing the entire equation by ...

So I'd divide [imath]4x^3-10x^2+2=0[/imath] by something to reduce the degree from 3 to 1. You multiplied it by x^2, increasing the degree to 5. And you multiplied one term of [imath]y=2/x^2 - x[/imath] by x^2, which is utterly wrong in the first place, beyond the fact that I didn't say to change "the latter" at all.

But thanks for showing your thoughts; that gives us a better idea of where you are going wrong. There's something very basic that you are missing.

 

[Kulla_9289]

Divide by x^2. So, 4x - 10 + 2/x^2 = 0. Then, minus x both sides. 4x - 10 + 2/x^2 - x= - x.
Then, 2x^2 = -x - 4x + 10. Hence, -5x + 10. Sorry for the mistake earlier; I overlooked many things.

@Dr.Peterson my question is that how would I know when to multiply or divide or add or subtract by a particular number or letter or so?

 

[Dr.Peterson]

Divide by x^2. So, 4x - 10 + 2/x^2 = 0. Then, minus x both sides. 4x - 10 + 2/x^2 - x= - x.
Then, 2x^2 = -x - 4x + 10. Hence, -5x + 10. Sorry for the mistake earlier; I overlooked many things.

I think you meant 2x^2 - x = -x - 4x + 10 = -5x + 10. That's good. (My thinking was just a little different, but only in detail.)

@Dr.Peterson my question is that how would I know when to multiply or divide or add or subtract by a particular number or letter or so?

Only by trying things! There is no general method; in fact, if you were given a random equation and a random graph, there would be no way to use the latter to solve the former. That you can depends on the fact that they chose the graph in some way so that they are related. Your task is to hold them both up in front of you (figuratively), turn them around in different directions, and find the connection.

"Turning it around" in this case meant things like looking at the degrees of terms (that is, looking for differences) and thinking about possible ways to make them similar. In other problems, that might take a different form.

As I've said repeatedly, this is the secret of problem solving: to be willing to try things without knowing what will work. Explore! Play! Discover! Too many students never learn that math can be an adventure; your textbook appears to be designed to teach you that, but you will only learn that as you follow their lead and try things. Don't expect every problem to be a familiar one. Expect to have to explore, and then do so.