Solve x-7/2 + 6/x = 1/2

[richiesmasher]

Hello, I have an equation here:

Solve (x-7)/2 + 6/x = 1/2

Now I've simplified as follows:

{x(x-7) + (6)(2)}/(2x) = 1/2

{x^2-7x +12}/(2x) = 1/2

Now I cross multiply and get
2(x^2-7x+12) = 2x

after factorizing the left hand side, I get

2 { (x-3) (x-4)} = 2x

and now I'm completely stuck, I'm trying to figure out what to do, should I have not factorized and brought across the 2x to make it a proper quadratic equation to use the formula to solve idk , someone please help

 

[Dr.Peterson]

Hello, I have an equation here:

Solve (x-7)/2 + 6/x = 1/2 <--- I assume you meant this

Now I've simplified as follows:

[x(x-7) + (6)(2)]/2x = 1/2 <--- I think you meant this

[x^2-7x +12]/2x = 1/2

Now I cross multiply and get
2(x^2-7x+12) = 2x

after factorizing the left hand side, I get

2 { (x-3) (x-4)} = 2x

and now I'm completely stuck, I'm trying to figure out what to do, should I have not factorized and brought across the 2x to make it a proper quadratic equation to use the formula to solve idk , someone please help

You shouldn't have factorized until you had a zero on the right (standard form for a quadratic equation). Factorization is useful only then. I would also divide both sides by 2 before factoring. (Or you could have just multiplied both sides by 2x instead of cross-multiplying, which often makes things more complicated than they have to be.)

 

[richiesmasher]

You shouldn't have factorized until you had a zero on the right (standard form for a quadratic equation). Factorization is useful only then. I would also divide both sides by 2 before factoring. (Or you could have just multiplied both sides by 2x instead of cross-multiplying, which often makes things more complicated than they have to be.)

Ok I understand what you're saying about multiplying both sides by 2x, but now I have x^2-7x+12 =2x/2 and I'm still unsure how to proceed from this point.

EDIT, I've actually solved it by multiplying both sides by 2, and resulting in

2x^2-14x+24 = 2x
2x^2 - 16x +24 = 0
from here I factorized to get
(2x-12) (x-2) = 0
and solved
x= 2
x=6

I don't know why i suddenly decided to try that but it worked, it seemd the most sensible to me, so simplify it to it's fullest.

 

[richiesmasher]

Easier if you start this way:

6/x = 1/2 - (x-7)/2
6/x = (1 - x + 7)/2

Got it?

Ah yes I see, that's a unique way,

However I'll end up cross multiplying either way, to get 12 = x-x^2+7x

From there I can solve it.

 

[lookagain]

.
.
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Now I've simplified as follows:

{x(x-7) + (6)(2)}/2x = 1/2

{x^2-7x +12}/2x = 1/2
.
.
.

These particular denominators need grouping symbols:

{x(x - 7) + (6)(2)}/(2x) = 1/2

{x^2 - 7x + 12}/(2x) = 1/2

 

[mmm4444bot]

Ok I understand what you're saying about multiplying both sides by 2x, but now I have x^2-7x+12 =2x/2 and I'm still unsure how to proceed from this point.

:idea: If you had simplified the right-hand side of your result above (i.e., the 2s cancel), perhaps you would have seen the next step: subtract x from each side.