- #1

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## Main Question or Discussion Point

Consider this equation:

[tex]7x*tan(4x)=5x[/tex]

Initially, one might think to simply factor the x out of each side of the equation and be left with this:

[tex]tan(4x)=\frac{5}{7}[/tex]

Well that's wrong! That is only one solution! The other solution is:

[tex]x=0[/tex]

You should instead have done the following:

[tex]7x*tan(4x)=5x[/tex]

[tex]7x*tan(4x)-5x=0[/tex]

[tex]x(7tan(4x)-5)=0[/tex]

[tex]x=0\ OR\ tan(4x)=\frac{5}{7}[/tex]

Now my question is: What properties of the initial equation should give me pause and should cause me to supress my reflex to factor the x out of the equation? Although in hindsight it is obvious that plugging in "0" for x would be a solution, that doesn't explain why doing a seemingly allowed operation (dividing each side by 7x) would withhold a solution from you.

[tex]7x*tan(4x)=5x[/tex]

Initially, one might think to simply factor the x out of each side of the equation and be left with this:

[tex]tan(4x)=\frac{5}{7}[/tex]

Well that's wrong! That is only one solution! The other solution is:

[tex]x=0[/tex]

You should instead have done the following:

[tex]7x*tan(4x)=5x[/tex]

[tex]7x*tan(4x)-5x=0[/tex]

[tex]x(7tan(4x)-5)=0[/tex]

[tex]x=0\ OR\ tan(4x)=\frac{5}{7}[/tex]

Now my question is: What properties of the initial equation should give me pause and should cause me to supress my reflex to factor the x out of the equation? Although in hindsight it is obvious that plugging in "0" for x would be a solution, that doesn't explain why doing a seemingly allowed operation (dividing each side by 7x) would withhold a solution from you.

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