confused about this c2 trig question

so the question is solve (1+cosx)^2= 1/4 for -180<x<180

I know that what you would have to do square root both sides and go on from there. but i did something different which should give the same answer but didn't

I squared it so i did (1+cosx)(1+cosx) to give me 1+2cosx+cos^2x=1/4

then i took the one to the other side to give me 2cosx+cos^2x=-3/4

then i factorised it to give me cosx(2+cosx)= -3/4

so cosx=-3/4 and 2+cosx=-3/4

doing x=cos^-1(-3/4) gives me 138.5903779
and doing x=cos^-1(-11/4) gives me a maths error

i went on further with the first one which gave me a solution but in fact the answer is +-120, i understand how that answer was obtained as you simply square root both sides and go on from there, but why was the method that i did (which should have given me the right answer) faulty?

You would have been correct had you taken the -3/4 back to the other side - you should then be able to see that this looks like a normal quadratic equation in cosx.

I think your method is faulty because what you have done isn't actually factorising - you could try saying "let cosx = t" to make it easier to see the equation as a more normal looking quadratic.

tHis is the quickest way square root both sides(1+cosx)^2=1/4 so 1 + cosx =sq rt of 1/4 which is 1/2 then minus 1 from both sides leaving u with cosx = -1/2 the do the inverse arcos-1/2 which gives u 120+-

Fully factorizing one side of the equation only works if the other side is 0.

This is because the step you are taking is:

$\alpha\beta\gamma=0 \Leftrightarrow \alpha=0\ \text{OR}\ \beta=0\ \text{OR}\ \gamma=0$

This is because of the Multiplicative Property of Zero (things multiplied together always equal zero iff at least one of them is zero).

oh i never knew that, and that applies for everything?

Fully factorizing one side of the equation only works if the other side is 0.

This is because the step you are taking is:

$\alpha\beta\gamma=0 \Leftrightarrow \alpha=0\ \text{OR}\ \beta=0\ \text{OR}\ \gamma=0$

This is because of the Multiplicative Property of Zero (things multiplied together always equal zero iff at least one of them is zero).

Aaah that's really interesting i thought of that but didn't know how to explain it properly

To be fair, Blazy said it before me, and I think PAAA was hinting at it ^^

I'm not quite sure why you expanded that whole thing:

(1+cosx)^2= 1/4

Square root both sides.

so the question is solve (1+cosx)^2= 1/4 for -180<x<180

I know that what you would have to do square root both sides and go on from there. but i did something different which should give the same answer but didn't

I squared it so i did (1+cosx)(1+cosx) to give me 1+2cosx+cos^2x=1/4

then i took the one to the other side to give me 2cosx+cos^2x=-3/4

then i factorised it to give me cosx(2+cosx)= -3/4

so cosx=-3/4 and 2+cosx=-3/4

doing x=cos^-1(-3/4) gives me 138.5903779
and doing x=cos^-1(-11/4) gives me a maths error

i went on further with the first one which gave me a solution but in fact the answer is +-120, i understand how that answer was obtained as you simply square root both sides and go on from there, but why was the method that i did (which should have given me the right answer) faulty?

oh i never knew that, and that applies for everything?