(2x-3)(3x-2)=90

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Solution for (2x-3)(3x-2)=90 equation:


Simplifying
(2x + -3)(3x + -2) = 90

Reorder the terms:
(-3 + 2x)(3x + -2) = 90

Reorder the terms:
(-3 + 2x)(-2 + 3x) = 90

Multiply (-3 + 2x) * (-2 + 3x)
(-3(-2 + 3x) + 2x * (-2 + 3x)) = 90
((-2 * -3 + 3x * -3) + 2x * (-2 + 3x)) = 90
((6 + -9x) + 2x * (-2 + 3x)) = 90
(6 + -9x + (-2 * 2x + 3x * 2x)) = 90
(6 + -9x + (-4x + 6x2)) = 90

Combine like terms: -9x + -4x = -13x
(6 + -13x + 6x2) = 90

Solving
6 + -13x + 6x2 = 90

Solving for variable 'x'.

Reorder the terms:
6 + -90 + -13x + 6x2 = 90 + -90

Combine like terms: 6 + -90 = -84
-84 + -13x + 6x2 = 90 + -90

Combine like terms: 90 + -90 = 0
-84 + -13x + 6x2 = 0

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

Divide each side by '6'.
-14 + -2.166666667x + x2 = 0

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + -2.166666667x + 14 + x2 = 0 + 14

Reorder the terms:
-14 + 14 + -2.166666667x + x2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + -2.166666667x + x2 = 0 + 14
-2.166666667x + x2 = 0 + 14

Combine like terms: 0 + 14 = 14
-2.166666667x + x2 = 14

The x term is -2.166666667x.  Take half its coefficient (-1.083333334).
Square it (1.173611113) and add it to both sides.

Add '1.173611113' to each side of the equation.
-2.166666667x + 1.173611113 + x2 = 14 + 1.173611113

Reorder the terms:
1.173611113 + -2.166666667x + x2 = 14 + 1.173611113

Combine like terms: 14 + 1.173611113 = 15.173611113
1.173611113 + -2.166666667x + x2 = 15.173611113

Factor a perfect square on the left side:
(x + -1.083333334)(x + -1.083333334) = 15.173611113

Calculate the square root of the right side: 3.895331964

Break this problem into two subproblems by setting 
(x + -1.083333334) equal to 3.895331964 and -3.895331964.

Subproblem 1

x + -1.083333334 = 3.895331964 Simplifying x + -1.083333334 = 3.895331964 Reorder the terms: -1.083333334 + x = 3.895331964 Solving -1.083333334 + x = 3.895331964 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + x = 3.895331964 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + x = 3.895331964 + 1.083333334 x = 3.895331964 + 1.083333334 Combine like terms: 3.895331964 + 1.083333334 = 4.978665298 x = 4.978665298 Simplifying x = 4.978665298

Subproblem 2

x + -1.083333334 = -3.895331964 Simplifying x + -1.083333334 = -3.895331964 Reorder the terms: -1.083333334 + x = -3.895331964 Solving -1.083333334 + x = -3.895331964 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + x = -3.895331964 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + x = -3.895331964 + 1.083333334 x = -3.895331964 + 1.083333334 Combine like terms: -3.895331964 + 1.083333334 = -2.81199863 x = -2.81199863 Simplifying x = -2.81199863

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.978665298, -2.81199863}

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