(3x+4)(2x+1)=180

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Solution for (3x+4)(2x+1)=180 equation:


Simplifying
(3x + 4)(2x + 1) = 180

Reorder the terms:
(4 + 3x)(2x + 1) = 180

Reorder the terms:
(4 + 3x)(1 + 2x) = 180

Multiply (4 + 3x) * (1 + 2x)
(4(1 + 2x) + 3x * (1 + 2x)) = 180
((1 * 4 + 2x * 4) + 3x * (1 + 2x)) = 180
((4 + 8x) + 3x * (1 + 2x)) = 180
(4 + 8x + (1 * 3x + 2x * 3x)) = 180
(4 + 8x + (3x + 6x2)) = 180

Combine like terms: 8x + 3x = 11x
(4 + 11x + 6x2) = 180

Solving
4 + 11x + 6x2 = 180

Solving for variable 'x'.

Reorder the terms:
4 + -180 + 11x + 6x2 = 180 + -180

Combine like terms: 4 + -180 = -176
-176 + 11x + 6x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-176 + 11x + 6x2 = 0

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

Divide each side by '6'.
-29.33333333 + 1.833333333x + x2 = 0

Move the constant term to the right:

Add '29.33333333' to each side of the equation.
-29.33333333 + 1.833333333x + 29.33333333 + x2 = 0 + 29.33333333

Reorder the terms:
-29.33333333 + 29.33333333 + 1.833333333x + x2 = 0 + 29.33333333

Combine like terms: -29.33333333 + 29.33333333 = 0.00000000
0.00000000 + 1.833333333x + x2 = 0 + 29.33333333
1.833333333x + x2 = 0 + 29.33333333

Combine like terms: 0 + 29.33333333 = 29.33333333
1.833333333x + x2 = 29.33333333

The x term is 1.833333333x.  Take half its coefficient (0.9166666665).
Square it (0.8402777775) and add it to both sides.

Add '0.8402777775' to each side of the equation.
1.833333333x + 0.8402777775 + x2 = 29.33333333 + 0.8402777775

Reorder the terms:
0.8402777775 + 1.833333333x + x2 = 29.33333333 + 0.8402777775

Combine like terms: 29.33333333 + 0.8402777775 = 30.1736111075
0.8402777775 + 1.833333333x + x2 = 30.1736111075

Factor a perfect square on the left side:
(x + 0.9166666665)(x + 0.9166666665) = 30.1736111075

Calculate the square root of the right side: 5.493051166

Break this problem into two subproblems by setting 
(x + 0.9166666665) equal to 5.493051166 and -5.493051166.

Subproblem 1

x + 0.9166666665 = 5.493051166 Simplifying x + 0.9166666665 = 5.493051166 Reorder the terms: 0.9166666665 + x = 5.493051166 Solving 0.9166666665 + x = 5.493051166 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + x = 5.493051166 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + x = 5.493051166 + -0.9166666665 x = 5.493051166 + -0.9166666665 Combine like terms: 5.493051166 + -0.9166666665 = 4.5763844995 x = 4.5763844995 Simplifying x = 4.5763844995

Subproblem 2

x + 0.9166666665 = -5.493051166 Simplifying x + 0.9166666665 = -5.493051166 Reorder the terms: 0.9166666665 + x = -5.493051166 Solving 0.9166666665 + x = -5.493051166 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + x = -5.493051166 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + x = -5.493051166 + -0.9166666665 x = -5.493051166 + -0.9166666665 Combine like terms: -5.493051166 + -0.9166666665 = -6.4097178325 x = -6.4097178325 Simplifying x = -6.4097178325

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.5763844995, -6.4097178325}

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