7(4x+1)(3x+2)=500

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Solution for 7(4x+1)(3x+2)=500 equation:


Simplifying
7(4x + 1)(3x + 2) = 500

Reorder the terms:
7(1 + 4x)(3x + 2) = 500

Reorder the terms:
7(1 + 4x)(2 + 3x) = 500

Multiply (1 + 4x) * (2 + 3x)
7(1(2 + 3x) + 4x * (2 + 3x)) = 500
7((2 * 1 + 3x * 1) + 4x * (2 + 3x)) = 500
7((2 + 3x) + 4x * (2 + 3x)) = 500
7(2 + 3x + (2 * 4x + 3x * 4x)) = 500
7(2 + 3x + (8x + 12x2)) = 500

Combine like terms: 3x + 8x = 11x
7(2 + 11x + 12x2) = 500
(2 * 7 + 11x * 7 + 12x2 * 7) = 500
(14 + 77x + 84x2) = 500

Solving
14 + 77x + 84x2 = 500

Solving for variable 'x'.

Reorder the terms:
14 + -500 + 77x + 84x2 = 500 + -500

Combine like terms: 14 + -500 = -486
-486 + 77x + 84x2 = 500 + -500

Combine like terms: 500 + -500 = 0
-486 + 77x + 84x2 = 0

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

Divide each side by '84'.
-5.785714286 + 0.9166666667x + x2 = 0

Move the constant term to the right:

Add '5.785714286' to each side of the equation.
-5.785714286 + 0.9166666667x + 5.785714286 + x2 = 0 + 5.785714286

Reorder the terms:
-5.785714286 + 5.785714286 + 0.9166666667x + x2 = 0 + 5.785714286

Combine like terms: -5.785714286 + 5.785714286 = 0.000000000
0.000000000 + 0.9166666667x + x2 = 0 + 5.785714286
0.9166666667x + x2 = 0 + 5.785714286

Combine like terms: 0 + 5.785714286 = 5.785714286
0.9166666667x + x2 = 5.785714286

The x term is 0.9166666667x.  Take half its coefficient (0.4583333334).
Square it (0.2100694445) and add it to both sides.

Add '0.2100694445' to each side of the equation.
0.9166666667x + 0.2100694445 + x2 = 5.785714286 + 0.2100694445

Reorder the terms:
0.2100694445 + 0.9166666667x + x2 = 5.785714286 + 0.2100694445

Combine like terms: 5.785714286 + 0.2100694445 = 5.9957837305
0.2100694445 + 0.9166666667x + x2 = 5.9957837305

Factor a perfect square on the left side:
(x + 0.4583333334)(x + 0.4583333334) = 5.9957837305

Calculate the square root of the right side: 2.448628949

Break this problem into two subproblems by setting 
(x + 0.4583333334) equal to 2.448628949 and -2.448628949.

Subproblem 1

x + 0.4583333334 = 2.448628949 Simplifying x + 0.4583333334 = 2.448628949 Reorder the terms: 0.4583333334 + x = 2.448628949 Solving 0.4583333334 + x = 2.448628949 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4583333334' to each side of the equation. 0.4583333334 + -0.4583333334 + x = 2.448628949 + -0.4583333334 Combine like terms: 0.4583333334 + -0.4583333334 = 0.0000000000 0.0000000000 + x = 2.448628949 + -0.4583333334 x = 2.448628949 + -0.4583333334 Combine like terms: 2.448628949 + -0.4583333334 = 1.9902956156 x = 1.9902956156 Simplifying x = 1.9902956156

Subproblem 2

x + 0.4583333334 = -2.448628949 Simplifying x + 0.4583333334 = -2.448628949 Reorder the terms: 0.4583333334 + x = -2.448628949 Solving 0.4583333334 + x = -2.448628949 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4583333334' to each side of the equation. 0.4583333334 + -0.4583333334 + x = -2.448628949 + -0.4583333334 Combine like terms: 0.4583333334 + -0.4583333334 = 0.0000000000 0.0000000000 + x = -2.448628949 + -0.4583333334 x = -2.448628949 + -0.4583333334 Combine like terms: -2.448628949 + -0.4583333334 = -2.9069622824 x = -2.9069622824 Simplifying x = -2.9069622824

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.9902956156, -2.9069622824}

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