x=(-2x-3)(3x-7)

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Solution for x=(-2x-3)(3x-7) equation:


Simplifying
x = (-2x + -3)(3x + -7)

Reorder the terms:
x = (-3 + -2x)(3x + -7)

Reorder the terms:
x = (-3 + -2x)(-7 + 3x)

Multiply (-3 + -2x) * (-7 + 3x)
x = (-3(-7 + 3x) + -2x * (-7 + 3x))
x = ((-7 * -3 + 3x * -3) + -2x * (-7 + 3x))
x = ((21 + -9x) + -2x * (-7 + 3x))
x = (21 + -9x + (-7 * -2x + 3x * -2x))
x = (21 + -9x + (14x + -6x2))

Combine like terms: -9x + 14x = 5x
x = (21 + 5x + -6x2)

Solving
x = 21 + 5x + -6x2

Solving for variable 'x'.

Reorder the terms:
-21 + x + -5x + 6x2 = 21 + 5x + -6x2 + -21 + -5x + 6x2

Combine like terms: x + -5x = -4x
-21 + -4x + 6x2 = 21 + 5x + -6x2 + -21 + -5x + 6x2

Reorder the terms:
-21 + -4x + 6x2 = 21 + -21 + 5x + -5x + -6x2 + 6x2

Combine like terms: 21 + -21 = 0
-21 + -4x + 6x2 = 0 + 5x + -5x + -6x2 + 6x2
-21 + -4x + 6x2 = 5x + -5x + -6x2 + 6x2

Combine like terms: 5x + -5x = 0
-21 + -4x + 6x2 = 0 + -6x2 + 6x2
-21 + -4x + 6x2 = -6x2 + 6x2

Combine like terms: -6x2 + 6x2 = 0
-21 + -4x + 6x2 = 0

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

Divide each side by '6'.
-3.5 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + -0.6666666667x + 3.5 + x2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + -0.6666666667x + x2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + -0.6666666667x + x2 = 0 + 3.5
-0.6666666667x + x2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
-0.6666666667x + x2 = 3.5

The x term is -0.6666666667x.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667x + 0.1111111112 + x2 = 3.5 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 3.5 + 0.1111111112

Combine like terms: 3.5 + 0.1111111112 = 3.6111111112
0.1111111112 + -0.6666666667x + x2 = 3.6111111112

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 3.6111111112

Calculate the square root of the right side: 1.900292375

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 1.900292375 and -1.900292375.

Subproblem 1

x + -0.3333333334 = 1.900292375 Simplifying x + -0.3333333334 = 1.900292375 Reorder the terms: -0.3333333334 + x = 1.900292375 Solving -0.3333333334 + x = 1.900292375 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = 1.900292375 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = 1.900292375 + 0.3333333334 x = 1.900292375 + 0.3333333334 Combine like terms: 1.900292375 + 0.3333333334 = 2.2336257084 x = 2.2336257084 Simplifying x = 2.2336257084

Subproblem 2

x + -0.3333333334 = -1.900292375 Simplifying x + -0.3333333334 = -1.900292375 Reorder the terms: -0.3333333334 + x = -1.900292375 Solving -0.3333333334 + x = -1.900292375 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = -1.900292375 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = -1.900292375 + 0.3333333334 x = -1.900292375 + 0.3333333334 Combine like terms: -1.900292375 + 0.3333333334 = -1.5669590416 x = -1.5669590416 Simplifying x = -1.5669590416

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.2336257084, -1.5669590416}

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