Avoiding Lost Solutions in Equation Solving

[member 529879]

When you have the equation (x)(x+1)=0 solving will give you x=0 or x=-1. If you divide both sides by x, you get the equation x+1=0. Solving this equation gives you only x=-1. Why was a solution lost, and how can that be avoided while solving other problems?

 

[Mentallic]

When you have the equation (x)(x+1)=0 solving will give you x=0 or x=-1. If you divide both sides by x, you get the equation x+1=0. Solving this equation gives you only x=-1. Why was a solution lost, and how can that be avoided while solving other problems?

When you divide by a variable, you're assuming that variable is not equal to 0. If in fact, one of the solutions is 0, then you've just lost it because of your assumption with dividing by zero. To avoid this problem, do not divide by variables, but rather factorize.

A lot of students when asked to solve
$$x^2+x=0$$
would first divide through by x resulting in
$$x+1=0$$
and then solve that. Do not do this, rather factorize it into
$$x(x+1)=0$$
and then solve each factor separately.

 

[chingel]

It's clear that when you have an expression with factors like $x(x+1)=0$, either of the factors can be zero and it's a solution to the equation, but if you get rid of one of the factors, you are no longer considering the solution where that factor would be zero.

Another way to think is that the results from division only make sense if you don't divide by zero. So the equation you get only makes sense if you didn't divide by zero and you have to separately consider what if the factor was zero.

 

[Mark44]

It's clear that when you have an expression with factors like $x(x+1)=0$, either of the factors can be zero and it's a solution to the equation, but if you get rid of one of the factors, you are no longer considering the solution where that factor would be zero.

Yes, exactly.

Another way to think is that the results from division only make sense if you don't divide by zero. So the equation you get only makes sense if you didn't divide by zero and you have to separately consider what if the factor was zero.

Right.

BTW, to write something in LaTeX here (on PhysicsForums), use two $ characters at the start, and two more at the end of your expression/equation, not just one. That renders the expression on its own line, centered in the page. For inline LaTeX, use two # characters at the start and two at the end. If you have just a simple equation, such as x(x + 1) = 0, I don't see much advantage in using LaTeX, but it's useful for exponents, square roots (and cube and higher roots), limits, integrals, fractions, and quite a few things more.