12x^2-18x-21=0

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Solution for 12x^2-18x-21=0 equation:


Simplifying
12x2 + -18x + -21 = 0

Reorder the terms:
-21 + -18x + 12x2 = 0

Solving
-21 + -18x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-7 + -6x + 4x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-7 + -6x + 4x2)' equal to zero and attempt to solve:

Simplifying
-7 + -6x + 4x2 = 0

Solving
-7 + -6x + 4x2 = 0

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

Divide each side by '4'.
-1.75 + -1.5x + x2 = 0

Move the constant term to the right:

Add '1.75' to each side of the equation.
-1.75 + -1.5x + 1.75 + x2 = 0 + 1.75

Reorder the terms:
-1.75 + 1.75 + -1.5x + x2 = 0 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + -1.5x + x2 = 0 + 1.75
-1.5x + x2 = 0 + 1.75

Combine like terms: 0 + 1.75 = 1.75
-1.5x + x2 = 1.75

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 1.75 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 1.75 + 0.5625

Combine like terms: 1.75 + 0.5625 = 2.3125
0.5625 + -1.5x + x2 = 2.3125

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 2.3125

Calculate the square root of the right side: 1.520690633

Break this problem into two subproblems by setting 
(x + -0.75) equal to 1.520690633 and -1.520690633.


Subproblem 1
x + -0.75 = 1.520690633

Simplifying
x + -0.75 = 1.520690633

Reorder the terms:
-0.75 + x = 1.520690633

Solving
-0.75 + x = 1.520690633

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = 1.520690633 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = 1.520690633 + 0.75
x = 1.520690633 + 0.75

Combine like terms: 1.520690633 + 0.75 = 2.270690633
x = 2.270690633

Simplifying
x = 2.270690633


Subproblem 2
x + -0.75 = -1.520690633

Simplifying
x + -0.75 = -1.520690633

Reorder the terms:
-0.75 + x = -1.520690633

Solving
-0.75 + x = -1.520690633

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = -1.520690633 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = -1.520690633 + 0.75
x = -1.520690633 + 0.75

Combine like terms: -1.520690633 + 0.75 = -0.770690633
x = -0.770690633

Simplifying
x = -0.770690633


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.270690633, -0.770690633}
Solution
x = {2.270690633, -0.770690633}

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