12x^2-28x-7=0

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Solution for 12x^2-28x-7=0 equation: 


Simplifying
12x2 + -28x + -7 = 0

Reorder the terms:
-7 + -28x + 12x2 = 0

Solving
-7 + -28x + 12x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
12 the coefficient of the squared term: 

Divide each side by '12'.
-0.5833333333 + -2.333333333x + x2 = 0

Move the constant term to the right:

Add '0.5833333333' to each side of the equation.
-0.5833333333 + -2.333333333x + 0.5833333333 + x2 = 0 + 0.5833333333

Reorder the terms:
-0.5833333333 + 0.5833333333 + -2.333333333x + x2 = 0 + 0.5833333333

Combine like terms: -0.5833333333 + 0.5833333333 = 0.0000000000
0.0000000000 + -2.333333333x + x2 = 0 + 0.5833333333
-2.333333333x + x2 = 0 + 0.5833333333

Combine like terms: 0 + 0.5833333333 = 0.5833333333
-2.333333333x + x2 = 0.5833333333

The x term is -2.333333333x.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333x + 1.361111112 + x2 = 0.5833333333 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333x + x2 = 0.5833333333 + 1.361111112

Combine like terms: 0.5833333333 + 1.361111112 = 1.9444444453
1.361111112 + -2.333333333x + x2 = 1.9444444453

Factor a perfect square on the left side:
(x + -1.166666667)(x + -1.166666667) = 1.9444444453

Calculate the square root of the right side: 1.394433378

Break this problem into two subproblems by setting 
(x + -1.166666667) equal to 1.394433378 and -1.394433378.


Subproblem 1
x + -1.166666667 = 1.394433378

Simplifying
x + -1.166666667 = 1.394433378

Reorder the terms:
-1.166666667 + x = 1.394433378

Solving
-1.166666667 + x = 1.394433378

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + x = 1.394433378 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + x = 1.394433378 + 1.166666667
x = 1.394433378 + 1.166666667

Combine like terms: 1.394433378 + 1.166666667 = 2.561100045
x = 2.561100045

Simplifying
x = 2.561100045


Subproblem 2
x + -1.166666667 = -1.394433378

Simplifying
x + -1.166666667 = -1.394433378

Reorder the terms:
-1.166666667 + x = -1.394433378

Solving
-1.166666667 + x = -1.394433378

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + x = -1.394433378 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + x = -1.394433378 + 1.166666667
x = -1.394433378 + 1.166666667

Combine like terms: -1.394433378 + 1.166666667 = -0.227766711
x = -0.227766711

Simplifying
x = -0.227766711


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.561100045, -0.227766711}

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