6x^2-9x-21=0

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Solution for 6x^2-9x-21=0 equation: 


Simplifying
6x2 + -9x + -21 = 0

Reorder the terms:
-21 + -9x + 6x2 = 0

Solving
-21 + -9x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-7 + -3x + 2x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-7 + -3x + 2x2)' equal to zero and attempt to solve:

Simplifying
-7 + -3x + 2x2 = 0

Solving
-7 + -3x + 2x2 = 0

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

Divide each side by '2'.
-3.5 + -1.5x + x2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + -1.5x + 3.5 + x2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + -1.5x + x2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + -1.5x + x2 = 0 + 3.5
-1.5x + x2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
-1.5x + x2 = 3.5

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 = 3.5 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 3.5 + 0.5625

Combine like terms: 3.5 + 0.5625 = 4.0625
0.5625 + -1.5x + x2 = 4.0625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 4.0625

Calculate the square root of the right side: 2.015564437

Break this problem into two subproblems by setting 
(x + -0.75) equal to 2.015564437 and -2.015564437.


Subproblem 1
x + -0.75 = 2.015564437

Simplifying
x + -0.75 = 2.015564437

Reorder the terms:
-0.75 + x = 2.015564437

Solving
-0.75 + x = 2.015564437

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 = 2.015564437 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = 2.015564437 + 0.75
x = 2.015564437 + 0.75

Combine like terms: 2.015564437 + 0.75 = 2.765564437
x = 2.765564437

Simplifying
x = 2.765564437


Subproblem 2
x + -0.75 = -2.015564437

Simplifying
x + -0.75 = -2.015564437

Reorder the terms:
-0.75 + x = -2.015564437

Solving
-0.75 + x = -2.015564437

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 = -2.015564437 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = -2.015564437 + 0.75
x = -2.015564437 + 0.75

Combine like terms: -2.015564437 + 0.75 = -1.265564437
x = -1.265564437

Simplifying
x = -1.265564437


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.765564437, -1.265564437}
Solution
x = {2.765564437, -1.265564437}

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