SOLUTION: solve the equation: 1/2(6+4x)-1/4(8x-12)=1/2(2x-4)

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Question 627563: solve the equation: 1/2(6+4x)-1/4(8x-12)=1/2(2x-4)
Answer by Maths68(1474) About Me  (Show Source):
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1%2F2%286%2B4x%29-1%2F4%288x-12%29=1%2F2%282x-4%29
1%2F%282%29%282%29%283%2B2x%29-1%2F%284%29%284%29%282x-3%29=1%2F%282%29%282%29%28x-2%29
1%2F4%283%2B2x%29-1%2F16%282x-3%29=1%2F4%28x-2%29
Multiply by 4 both sides
4%281%2F4%283%2B2x%29%29-4%281%2F16%282x-3%29%29=4%281%2F4%28x-2%29%29

1%2F%283%2B2x%29-1%2F4%282x-3%29=1%2F%28x-2%29
Take LCD
%284%282x-3%29-%283%2B2x%29%29%2F%284%283%2B2x%29%282x-3%29%29=1%2F%28x-2%29
%288x-12-3-2x%29%2F%284%283%2B2x%29%282x-3%29%29=1%2F%28x-2%29
%286x-15%29%2F%284%283%2B2x%29%282x-3%29%29=1%2F%28x-2%29
%286x-15%29%28x-2%29=%281%29%284%29%283%2B2x%29%282x-3%29
6x%5E2-12x-15x%2B30=4%286x-9%2B4x%5E2-6x%29
6x%5E2-27x%2B30=4%28cross%286x%29-9%2B4x%5E2-cross%286x%29%29
6x%5E2-27x%2B30=4%28-9%2B4x%5E2%29
6x%5E2-27x%2B30=-36%2B16x%5E2
6x%5E2-27x%2B30%2B36-16x%5E2=0
6x%5E2-16x%5E2-27x%2B30%2B36=0
-10x%5E2-27x%2B66=0
Multiply by -1
10x%5E2%2B27x-66=0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 10x%5E2%2B27x%2B-66+=+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=%2827%29%5E2-4%2A10%2A-66=3369.

Discriminant d=3369 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-27%2B-sqrt%28+3369+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2827%29%2Bsqrt%28+3369+%29%29%2F2%5C10+=+1.55215437218629
x%5B2%5D+=+%28-%2827%29-sqrt%28+3369+%29%29%2F2%5C10+=+-4.25215437218629

Quadratic expression 10x%5E2%2B27x%2B-66 can be factored:
10x%5E2%2B27x%2B-66+=+10%28x-1.55215437218629%29%2A%28x--4.25215437218629%29
Again, the answer is: 1.55215437218629, -4.25215437218629. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+10%2Ax%5E2%2B27%2Ax%2B-66+%29