SOLUTION: Please help me solve this: {{{(12x^2)/(3x^2+12x)}}} divided by {{{(4x^3-36x)/(x^2+x-12)}}}

Question 78331: Please help me solve this:
divided by
Found 2 solutions by jim_thompson5910, stanbon:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Start with the given expression

Multiply the 1st fraction by the reciprocal of the 2nd

Factor out an x from the denominator of the right term

Factor out an x from the denominator of the left term

Factor the numerator of the 1st term

Solved by pluggable solver: Factoring Quadratics with a leading coefficient of 1 (a=1)

In order to factor , first we need to ask ourselves: What two numbers multiply to -12 and add to 1? Lets find out by listing all of the possible factors of -12

Factors:

1,2,3,4,6,12,

-1,-2,-3,-4,-6,-12,List the negative factors as well. This will allow us to find all possible combinations

These factors pair up to multiply to -12.

(-1)*(12)=-12

(-2)*(6)=-12

(-3)*(4)=-12

Now which of these pairs add to 1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 1

||||||
First Number | Second Number | Sum
1 | -12 | 1+(-12)=-11
2 | -6 | 2+(-6)=-4
3 | -4 | 3+(-4)=-1
-1 | 12 | (-1)+12=11
-2 | 6 | (-2)+6=4
-3 | 4 | (-3)+4=1
We can see from the table that -3 and 4 add to 1.So the two numbers that multiply to -12 and add to 1 are: -3 and 4 Now we substitute these numbers into a and b of the general equation of a product of linear factors which is: substitute a=-3 and b=4 So the equation becomes: (x-3)(x+4) Notice that if we foil (x-3)(x+4) we get the quadratic again

Now factor the denominator of the 2nd term

ERROR Algebra::Solver::Engine::invoke_solver_noengine: solver not defined for name 'quadratic_factoring'.

Error occurred executing solver 'quadratic_factoring' .

Basically factors to:

Factor out a 2 out of

Notice these terms cancel

Factor a 3 out of

Notice these terms cancel

So we get after reducing:

Now multiply

Divide and simplify

Factor out a 2

Divide and simplify again. This is your simplified answer.
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
(12x^2)/(3x^2+12x) divided by (4x^3-36x)/(x^2+x-12)
Factor where you can:
(12x^2)/(3x(x+4) / 4x(x+3)(x-3) / (x+4)(x-3)
Cancel where you can to get: invert the denominator and multiply
(4x)/(x+4) * (x+4)/4x(x+3)
Cancel where you can to get:
= 1/(x+3)
===========
Cheers,
Stan H.
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