x^2+3x-2=4x^2-5x-1

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Solution for x^2+3x-2=4x^2-5x-1 equation: 


Simplifying
x2 + 3x + -2 = 4x2 + -5x + -1

Reorder the terms:
-2 + 3x + x2 = 4x2 + -5x + -1

Reorder the terms:
-2 + 3x + x2 = -1 + -5x + 4x2

Solving
-2 + 3x + x2 = -1 + -5x + 4x2

Solving for variable 'x'.

Reorder the terms:
-2 + 1 + 3x + 5x + x2 + -4x2 = -1 + -5x + 4x2 + 1 + 5x + -4x2

Combine like terms: -2 + 1 = -1
-1 + 3x + 5x + x2 + -4x2 = -1 + -5x + 4x2 + 1 + 5x + -4x2

Combine like terms: 3x + 5x = 8x
-1 + 8x + x2 + -4x2 = -1 + -5x + 4x2 + 1 + 5x + -4x2

Combine like terms: x2 + -4x2 = -3x2
-1 + 8x + -3x2 = -1 + -5x + 4x2 + 1 + 5x + -4x2

Reorder the terms:
-1 + 8x + -3x2 = -1 + 1 + -5x + 5x + 4x2 + -4x2

Combine like terms: -1 + 1 = 0
-1 + 8x + -3x2 = 0 + -5x + 5x + 4x2 + -4x2
-1 + 8x + -3x2 = -5x + 5x + 4x2 + -4x2

Combine like terms: -5x + 5x = 0
-1 + 8x + -3x2 = 0 + 4x2 + -4x2
-1 + 8x + -3x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
-1 + 8x + -3x2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
0.3333333333 + -2.666666667x + x2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -2.666666667x + -0.3333333333 + x2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -2.666666667x + x2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -2.666666667x + x2 = 0 + -0.3333333333
-2.666666667x + x2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-2.666666667x + x2 = -0.3333333333

The x term is -2.666666667x.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667x + 1.777777780 + x2 = -0.3333333333 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x + x2 = -0.3333333333 + 1.777777780

Combine like terms: -0.3333333333 + 1.777777780 = 1.4444444467
1.777777780 + -2.666666667x + x2 = 1.4444444467

Factor a perfect square on the left side:
(x + -1.333333334)(x + -1.333333334) = 1.4444444467

Calculate the square root of the right side: 1.201850426

Break this problem into two subproblems by setting 
(x + -1.333333334) equal to 1.201850426 and -1.201850426.


Subproblem 1
x + -1.333333334 = 1.201850426

Simplifying
x + -1.333333334 = 1.201850426

Reorder the terms:
-1.333333334 + x = 1.201850426

Solving
-1.333333334 + x = 1.201850426

Solving for variable 'x'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + x = 1.201850426 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = 1.201850426 + 1.333333334
x = 1.201850426 + 1.333333334

Combine like terms: 1.201850426 + 1.333333334 = 2.53518376
x = 2.53518376

Simplifying
x = 2.53518376


Subproblem 2
x + -1.333333334 = -1.201850426

Simplifying
x + -1.333333334 = -1.201850426

Reorder the terms:
-1.333333334 + x = -1.201850426

Solving
-1.333333334 + x = -1.201850426

Solving for variable 'x'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + x = -1.201850426 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = -1.201850426 + 1.333333334
x = -1.201850426 + 1.333333334

Combine like terms: -1.201850426 + 1.333333334 = 0.131482908
x = 0.131482908

Simplifying
x = 0.131482908


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.53518376, 0.131482908}

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