SOLUTION: What are the factors for the product of -36 and the sum of 12? The original problem was 2q/2q+3 -2q/2q-3=1

Algebra ->  Rational-functions -> SOLUTION: What are the factors for the product of -36 and the sum of 12? The original problem was 2q/2q+3 -2q/2q-3=1      Log On


   

Question 809383: What are the factors for the product of -36 and the sum of 12?
The original problem was
2q/2q+3 -2q/2q-3=1
Found 2 solutions by edjones, KMST:
Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
2q/2q+3 - 2q/2q-3 = 1
4q^2-6q-4q^2-6q=4q^2-9
4q^2+12q-9=0
q=-3(sqrt2+1)/2, q=3(sqrt2-1)/2 see below
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Ed
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Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B12x%2B-9+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2812%29%5E2-4%2A4%2A-9=288.

Discriminant d=288 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-12%2B-sqrt%28+288+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2812%29%2Bsqrt%28+288+%29%29%2F2%5C4+=+0.621320343559642
x%5B2%5D+=+%28-%2812%29-sqrt%28+288+%29%29%2F2%5C4+=+-3.62132034355964

Quadratic expression 4x%5E2%2B12x%2B-9 can be factored:
4x%5E2%2B12x%2B-9+=+4%28x-0.621320343559642%29%2A%28x--3.62132034355964%29
Again, the answer is: 0.621320343559642, -3.62132034355964. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B12%2Ax%2B-9+%29

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Not all quadratic equations can be solved by factoring.
Factoring works when the solutions are rational numbers.
That is not the case for 4q%5E2%2B12q-9=0 .
I would solve that by completing the square,
or by applying the quadratic formula.

Completing the square:
4q%5E2%2B12q-9=0 --> 4q%5E2%2B12q=9
Looking at 4q%5E2%2B12q , you realize that it is part of
2q%2B3%29%5E2=4q%5E2%2B12q%2B9 ,
so you add 9 to both sides of 4q%5E2%2B12q=9 to get
4q%5E2%2B12q%2B9=9%2B9 --> 2q%2B3%29%5E2=18
Either
2q%2B3=sqrt%2818%29-->2q%2B3=3sqrt%282%29-->2q=-3%2B3sqrt%282%29-->highlight%28q=%28-3%2B3sqrt%282%29%29%2F2%29
or
2q%2B3=-sqrt%2818%29-->2q%2B3=-3sqrt%282%29-->2q=-3-3sqrt%282%29-->highlight%28q=%28-3-3sqrt%282%29%29%2F2%29
Both solutions can bew expressed with the formula
highlight%28q=%28-3+%2B-+3sqrt%282%29%29%2F2%29

Applying the quadratic formula works well too, but it is kind of boring.
(I only use it when the square to be completed is not too obvious).
Assuming that you remember the formula and apply it flawlessly, you end up with