(2x+3)(3x-2)=6

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Solution for (2x+3)(3x-2)=6 equation:


Simplifying
(2x + 3)(3x + -2) = 6

Reorder the terms:
(3 + 2x)(3x + -2) = 6

Reorder the terms:
(3 + 2x)(-2 + 3x) = 6

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

Combine like terms: 9x + -4x = 5x
(-6 + 5x + 6x2) = 6

Solving
-6 + 5x + 6x2 = 6

Solving for variable 'x'.

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

Combine like terms: -6 + -6 = -12
-12 + 5x + 6x2 = 6 + -6

Combine like terms: 6 + -6 = 0
-12 + 5x + 6x2 = 0

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

Divide each side by '6'.
-2 + 0.8333333333x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 0.8333333333x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + 0.8333333333x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 0.8333333333x + x2 = 0 + 2
0.8333333333x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
0.8333333333x + x2 = 2

The x term is 0.8333333333x.  Take half its coefficient (0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
0.8333333333x + 0.1736111111 + x2 = 2 + 0.1736111111

Reorder the terms:
0.1736111111 + 0.8333333333x + x2 = 2 + 0.1736111111

Combine like terms: 2 + 0.1736111111 = 2.1736111111
0.1736111111 + 0.8333333333x + x2 = 2.1736111111

Factor a perfect square on the left side:
(x + 0.4166666667)(x + 0.4166666667) = 2.1736111111

Calculate the square root of the right side: 1.474317168

Break this problem into two subproblems by setting 
(x + 0.4166666667) equal to 1.474317168 and -1.474317168.

Subproblem 1

x + 0.4166666667 = 1.474317168 Simplifying x + 0.4166666667 = 1.474317168 Reorder the terms: 0.4166666667 + x = 1.474317168 Solving 0.4166666667 + x = 1.474317168 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = 1.474317168 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = 1.474317168 + -0.4166666667 x = 1.474317168 + -0.4166666667 Combine like terms: 1.474317168 + -0.4166666667 = 1.0576505013 x = 1.0576505013 Simplifying x = 1.0576505013

Subproblem 2

x + 0.4166666667 = -1.474317168 Simplifying x + 0.4166666667 = -1.474317168 Reorder the terms: 0.4166666667 + x = -1.474317168 Solving 0.4166666667 + x = -1.474317168 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = -1.474317168 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = -1.474317168 + -0.4166666667 x = -1.474317168 + -0.4166666667 Combine like terms: -1.474317168 + -0.4166666667 = -1.8909838347 x = -1.8909838347 Simplifying x = -1.8909838347

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.0576505013, -1.8909838347}

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