Finding X and Y intercepts

S

Southern Geologist

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Hello,

I'm working through the algebra/pre-calculus review section of a calculus textbook and came across a problem that has me flummoxed. I've looked up the solution here (Chapter P, Sec. 1, Ex. 23), but it's been of little use to me as some skips are stepped.

Here is the problem:

Y = 3(2 - √ X)
__________
​
X
​

Solving for the Y intercept is easy enough, it comes out undefined, but I cannot figure out how to factor that for the X intercept, and certainly not how to reduce it to 0 = 2 - √ X that is indicated by the CalcChat answer.

Any help is greatly appreciated.
 
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x-intercept MEANS y = 0

Looking at the expression on the right-hand side, we see that it is not defined at x = 0.

Having said that, how do we find x such that y = 0? 3 is not zero. We know x is not zero.

2 - sqrt(x) needs to be zero. Why are you trying to factor this?
 
The textbook recommended factoring, albeit on a different equation.

I understand that 2 - sqrt X needs to equal zero, I'm just not sure how to remove the rest of the equation (the 3 and the x on the bottom half of the fraction) to get to 2 - sqrt X = zero.
 
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I figured the problem out with the help of a friend. Multiply both sides by X to remove the denominator and everything falls into place afterwards. No more help is needed.
 
Why is it appropriate to "multiply by x"? Are you SURE it's okay to do that?
 
Am I sure it's okay to do that? Well, no, not completely, but it's the only solution that I and a few other people viewing the problem have been able to come up with that leads to the next step (2 - sqrt x = 0).
 
Truthfully, that "step" is unnecessary.

I addressed the issue in my original comments. If your problem were this, y = 2x(2-sqrt(x))/x, might it be any different?
 
Ahh....I think I'm beginning to understand. Rather than trying to simplify the equation, I should -- as you noted earlier and I didn't understand -- figure out which part of the equation can be set to zero, 'discard' the rest, and focus on that? I was unaware that that was allowed.
 
Very good. Always keep what I call "Domain Issues" in your head.

I would have solved the original problem using this thought process.

1) I want Y to be zero.
2) I don't like that denominator, so let's just point out that x cannot be zero.
3) '3' cannot be zero. No need to worry about that.
4) 2 - sqrt(x) is all we have left. Find where that is zero and we're done.

It's more of a thought progression than a process to be memorized. Really, VERY good work!

Note: Absolutely nothing wrong with just ploughing through the algebra, but let's make sure we know what we are doing along the way.
 
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Thank you very much for the assistance and for your patience. I'll try to keep that advice in my head and work off of similar principles in the future.

One of the primary defects of my math education (and what I'm currently using self-study to work on!) is that I was always taught to memorize things rather than making use of the underlying thought processes/logic and it's wonderful to see someone giving out advice here that aims at that kind of higher level of math education.
 
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