(2x-3)(3x+2)=4

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Solution for (2x-3)(3x+2)=4 equation:


Simplifying
(2x + -3)(3x + 2) = 4

Reorder the terms:
(-3 + 2x)(3x + 2) = 4

Reorder the terms:
(-3 + 2x)(2 + 3x) = 4

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

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

Solving
-6 + -5x + 6x2 = 4

Solving for variable 'x'.

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

Combine like terms: -6 + -4 = -10
-10 + -5x + 6x2 = 4 + -4

Combine like terms: 4 + -4 = 0
-10 + -5x + 6x2 = 0

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

Divide each side by '6'.
-1.666666667 + -0.8333333333x + x2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + -0.8333333333x + 1.666666667 + x2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + -0.8333333333x + x2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + -0.8333333333x + x2 = 0 + 1.666666667
-0.8333333333x + x2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
-0.8333333333x + x2 = 1.666666667

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 = 1.666666667 + 0.1736111111

Reorder the terms:
0.1736111111 + -0.8333333333x + x2 = 1.666666667 + 0.1736111111

Combine like terms: 1.666666667 + 0.1736111111 = 1.8402777781
0.1736111111 + -0.8333333333x + x2 = 1.8402777781

Factor a perfect square on the left side:
(x + -0.4166666667)(x + -0.4166666667) = 1.8402777781

Calculate the square root of the right side: 1.356568383

Break this problem into two subproblems by setting 
(x + -0.4166666667) equal to 1.356568383 and -1.356568383.

Subproblem 1

x + -0.4166666667 = 1.356568383 Simplifying x + -0.4166666667 = 1.356568383 Reorder the terms: -0.4166666667 + x = 1.356568383 Solving -0.4166666667 + x = 1.356568383 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 = 1.356568383 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = 1.356568383 + 0.4166666667 x = 1.356568383 + 0.4166666667 Combine like terms: 1.356568383 + 0.4166666667 = 1.7732350497 x = 1.7732350497 Simplifying x = 1.7732350497

Subproblem 2

x + -0.4166666667 = -1.356568383 Simplifying x + -0.4166666667 = -1.356568383 Reorder the terms: -0.4166666667 + x = -1.356568383 Solving -0.4166666667 + x = -1.356568383 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 = -1.356568383 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = -1.356568383 + 0.4166666667 x = -1.356568383 + 0.4166666667 Combine like terms: -1.356568383 + 0.4166666667 = -0.9399017163 x = -0.9399017163 Simplifying x = -0.9399017163

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.7732350497, -0.9399017163}

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