3x(2x-1)=21

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Solution for 3x(2x-1)=21 equation: 


Simplifying
3x(2x + -1) = 21

Reorder the terms:
3x(-1 + 2x) = 21
(-1 * 3x + 2x * 3x) = 21
(-3x + 6x2) = 21

Solving
-3x + 6x2 = 21

Solving for variable 'x'.

Reorder the terms:
-21 + -3x + 6x2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + -3x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-7 + -1x + 2x2) = 0

Ignore the factor 3.

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

Simplifying
-7 + -1x + 2x2 = 0

Solving
-7 + -1x + 2x2 = 0

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

Divide each side by '2'.
-3.5 + -0.5x + x2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + -0.5x + 3.5 + x2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + -0.5x + x2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + -0.5x + x2 = 0 + 3.5
-0.5x + x2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
-0.5x + x2 = 3.5

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 3.5 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 3.5 + 0.0625

Combine like terms: 3.5 + 0.0625 = 3.5625
0.0625 + -0.5x + x2 = 3.5625

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(x + -0.25) equal to 1.887458609 and -1.887458609.


Subproblem 1
x + -0.25 = 1.887458609

Simplifying
x + -0.25 = 1.887458609

Reorder the terms:
-0.25 + x = 1.887458609

Solving
-0.25 + x = 1.887458609

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x = 1.887458609 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = 1.887458609 + 0.25
x = 1.887458609 + 0.25

Combine like terms: 1.887458609 + 0.25 = 2.137458609
x = 2.137458609

Simplifying
x = 2.137458609


Subproblem 2
x + -0.25 = -1.887458609

Simplifying
x + -0.25 = -1.887458609

Reorder the terms:
-0.25 + x = -1.887458609

Solving
-0.25 + x = -1.887458609

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x = -1.887458609 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = -1.887458609 + 0.25
x = -1.887458609 + 0.25

Combine like terms: -1.887458609 + 0.25 = -1.637458609
x = -1.637458609

Simplifying
x = -1.637458609


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.137458609, -1.637458609}
Solution
x = {2.137458609, -1.637458609}

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