2x^2-2x-3=27

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Solution for 2x^2-2x-3=27 equation: 


Simplifying
2x2 + -2x + -3 = 27

Reorder the terms:
-3 + -2x + 2x2 = 27

Solving
-3 + -2x + 2x2 = 27

Solving for variable 'x'.

Reorder the terms:
-3 + -27 + -2x + 2x2 = 27 + -27

Combine like terms: -3 + -27 = -30
-30 + -2x + 2x2 = 27 + -27

Combine like terms: 27 + -27 = 0
-30 + -2x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-15 + -1x + x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-15 + -1x + x2)' equal to zero and attempt to solve:

Simplifying
-15 + -1x + x2 = 0

Solving
-15 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + -1x + 15 + x2 = 0 + 15

Reorder the terms:
-15 + 15 + -1x + x2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + -1x + x2 = 0 + 15
-1x + x2 = 0 + 15

Combine like terms: 0 + 15 = 15
-1x + x2 = 15

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 15 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 15 + 0.25

Combine like terms: 15 + 0.25 = 15.25
0.25 + -1x + x2 = 15.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 15.25

Calculate the square root of the right side: 3.905124838

Break this problem into two subproblems by setting 
(x + -0.5) equal to 3.905124838 and -3.905124838.


Subproblem 1
x + -0.5 = 3.905124838

Simplifying
x + -0.5 = 3.905124838

Reorder the terms:
-0.5 + x = 3.905124838

Solving
-0.5 + x = 3.905124838

Solving for variable 'x'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x = 3.905124838 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = 3.905124838 + 0.5
x = 3.905124838 + 0.5

Combine like terms: 3.905124838 + 0.5 = 4.405124838
x = 4.405124838

Simplifying
x = 4.405124838


Subproblem 2
x + -0.5 = -3.905124838

Simplifying
x + -0.5 = -3.905124838

Reorder the terms:
-0.5 + x = -3.905124838

Solving
-0.5 + x = -3.905124838

Solving for variable 'x'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x = -3.905124838 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = -3.905124838 + 0.5
x = -3.905124838 + 0.5

Combine like terms: -3.905124838 + 0.5 = -3.405124838
x = -3.405124838

Simplifying
x = -3.405124838


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.405124838, -3.405124838}
Solution
x = {4.405124838, -3.405124838}

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