I was working on an engineering problem where I had to find the value of a component. I used Maxima (a computer algebra system) to get my answer, except that I got two answers instead of one. Only one of the answers satisfied my initial conditions.

So my question is "when do you need to check your answers"? I seem to recall some stuff about symmetrical functions or always positive or something else. Can anyone help me out or point me to some reading?

If anyone wants to know the exact equations I was using, they are here.

L1*L2 - M^2
------------------- = Leq
L1+L2 - 2*M

where L1 = L2 = 0.2
and Leq = 0.14

---->Solve for M

The answer I got was M = 0.2 and M = 0.08

If you plug in M = 0.2 you get 0.

Can anyone explain where this false answer came from?

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If you plug in M = 0.2 you get 0.
$$L_1+L_2-2M=0$$ so the whole thing is undefined, not 0.

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$L_1+L_2-2M=0$ so the whole thing is undefined, not 0.

Ooops, that is true. But it still doesn't answer the main question. One of the answers is still invalid.

If the equation is to make any sense, we must assume that $$L_1+L_2\not=2M$$

If the equation is to make any sense, we must assume that $$L_1+L_2\not=2M$$
I know. That assumption is what made me know that one of the answers was bogus. But the question I am asking is how do you know when to expect bogus answers so you know to test them instead of just assuming they are correct?

If there are radicals involved in the equation to be solved, then there is always a chance that one of the answers doesn't work, and this must always be checked because it's hard to catch. When there are fractions, division by zero could occur, but you don't need to check it in most cases as long as you take not of the bogus values before you solve the problem. If you are about to divide by a variable which isn't necessarily zero, then a solution is often times lost. When you're working with functions and their inverses (such as trig functions), be aware of the domain and range of the function and check you're answer, unless the function is bijective.

If there are radicals involved in the equation to be solved, then there is always a chance that one of the answers doesn't work, and this must always be checked because it's hard to catch. When there are fractions, division by zero could occur, but you don't need to check it in most cases as long as you take not of the bogus values before you solve the problem. If you are about to divide by a variable which isn't necessarily zero, then a solution is often times lost. When you're working with functions and their inverses (such as trig functions), be aware of the domain and range of the function and check you're answer, unless the function is bijective.

D H
Staff Emeritus
So my question is "when do you need to check your answers"?
Nobody has answered this simple question yet. The answer is equally simple: Always.

Engineers and scientists working in the real world often make decisions that, if wrong, can cost their employers an immense amount of money. Graduates fresh out of college can cost their employer tens of thousands of dollars with a simple screwup. Nobody trusts freshouts, at least not initially. Once you have gained your employer's trust, your mistakes might cost them hundreds of thousands, then millions, and later on in your career, even more. Employers trust these people even less. Complex projects often have a special team, the verification and validation team, whose sole job is to detect mistakes. Extremely complex projects have yet another team, the independent verification and validation team, whose sole job is to detect mistakes that have slipped through all the checks and balances.

Just because multiple layers of people exist whose jobs are to detect errors made by the design and development teams does not give the designers and developers free reign to make mistakes. Those verification and validation jobs exist in small quantities because the original designers are expected to have done their due diligence in checking their own work. Sloppy work done by the design and development team inevitably leads to incomplete work done by the verification and validation teams and a garbage product going out the door.

Moral of the story: Get in the habit of always checking your work.

Nobody has answered this simple question yet. The answer is equally simple: Always.

Engineers and scientists working in the real world often make decisions that, if wrong, can cost their employers an immense amount of money. Graduates fresh out of college can cost their employer tens of thousands of dollars with a simple screwup. Nobody trusts freshouts, at least not initially. Once you have gained your employer's trust, your mistakes might cost them hundreds of thousands, then millions, and later on in your career, even more. Employers trust these people even less. Complex projects often have a special team, the verification and validation team, whose sole job is to detect mistakes. Extremely complex projects have yet another team, the independent verification and validation team, whose sole job is to detect mistakes that have slipped through all the checks and balances.

Just because multiple layers of people exist whose jobs are to detect errors made by the design and development teams does not give the designers and developers free reign to make mistakes. Those verification and validation jobs exist in small quantities because the original designers are expected to have done their due diligence in checking their own work. Sloppy work done by the design and development team inevitably leads to incomplete work done by the verification and validation teams and a garbage product going out the door.

Moral of the story: Get in the habit of always checking your work.

Thanks for that real world info. I think I should have been more specific about my question though. I didn't mean checking work for errors in the solutions, but knowing when bogus answers are possible. I guess getting more than one answer usually means you should check to make sure they both work, especially when only one answer should physically make sense.

As far as checking for the correct answer, I think I am covered because I spend almost as much time checking my answers on my homework as actually solve them. I usually run a simulation in spice (electrical engineering simulation tool) to make sure it checks out.

Redbelly98
Staff Emeritus
Homework Helper
It's a matter of learning which mathematical operations can introduce extraneous solutions.

If the derivation involves:

• squaring both sides of the equation (as is often done to eliminate radicals)
• raising both sides of the equation to any even power
• taking the sine, cosine, or any periodic function of the equation
• employing any function whose inverse operation is not itself a function (such as squaring, sine, cosh, etc.)

... then you need to check for false answers.

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It's a matter of learning which mathematical operations can introduce extraneous solutions.

If the derivation involves:

• squaring both sides of the equation (as is often done to eliminate radicals)
• raising both sides of the equation to any even power
• taking the sine, cosine, or any periodic function of the equation
• employing any function whose inverse operation is not itself a function (such as squaring, sine, cosh, etc.)

... then you need to check for false answers.
also, multiplying both sides of an equation by an expression containing a variable.