Simplify x+7/2x+12 add 6/x^2-36

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txmedteach's profile pic

txmedteach | High School Teacher | (Level 3) Associate Educator

Posted on

You actually end up factoring the numerator to get

`((x+6)(x-5))/(2(x+6)(x-6)) = (x-5)/(2(x-6))`

Not a big difference. But I guess the takeaway point here is to make sure you have the basics of factoring and combining "like" terms down, so you could trace what sciencesolve did and figure out what went wrong, if anything. You can't just say "It doesn't look right," without backing it up!

The best way to factor that I've learned is a bit different, but I've found it to be pretty helpful:

Suppose you're given an expression of the form:

`x^2 + ax + b `

You know that can be factored somehow into:


Now, if we expand this out:


And now we get it to look like our first expression:

`x^2 + (y+z)x + yz`

We can see that:

`a = y+z`

`b = yz`

Now, this basically means the coefficient of x is the sum (or difference if one is negative) of the constant part of your factors, and the constant term is their product.

So, here, you see the expression:


You can then think of "What two numbers add to 1, and multiply to negative 30?"

Bam: 6 and -5

Your factors are now (x+6) and (x-5).

Hope that helps. But yeah, something doesn't look right usually means that it isn't.

The only way to figure out if it is right? Go through the steps, figure out what's going on between each step, and if you can't see how one step led to the next, figure out what you think it should go to and see if the two are different. Trace the problem!

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to write the simplified form of addition `(x+7)/(2x+12) + 6/(x^2-36), ` hence you need to factor 2 out to denominator `2x + 12 ` such that:

`(x+7)/(2(x+6)) + 6/(x^2-36)`

`` You also need to substitute (x-6)(x+6) for the denominator x^2 - 36 such that:

`(x+7)/(2(x+6)) + 6/((x-6)(x+6))`

You need to bring the fractions to a common denominator such that you need to multiply first fractions by (x-6) and the second fraction by 2, hence:

`((x+7)(x-6) + 2*6)/(2(x+6)(x-6))`

Removing the brackets to numerator yields:

`(x^2 - 6x + 7x - 42 + 12)/(2(x+6)(x-6))`

`(x^2 + x- 30)/(2(x+6)(x-6))`

You need to factor the numerator such that:

`(x^2 + 30x - x - 30)/(2(x+6)(x-6)) = ((x^2 - x) + 30(x-1))/(2(x+6)(x-6))`

`` `((x^2 - x) + 30(x-1))/(2(x+6)(x-6)) = (x(x - 1) + 30(x-1))/(2(x+6)(x-6))`

You need to factor out (x-1) to numerator such that:


Hence, evaluating the addition of the fractions by bringing it to the simplest form yields `((x-1)(x+30))/(2(x+6)(x-6)).`

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