3(x+5)=2(2x+3)x-2

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Solution for 3(x+5)=2(2x+3)x-2 equation:


Simplifying
3(x + 5) = 2(2x + 3) * x + -2

Reorder the terms:
3(5 + x) = 2(2x + 3) * x + -2
(5 * 3 + x * 3) = 2(2x + 3) * x + -2
(15 + 3x) = 2(2x + 3) * x + -2

Reorder the terms:
15 + 3x = 2(3 + 2x) * x + -2

Reorder the terms for easier multiplication:
15 + 3x = 2x(3 + 2x) + -2
15 + 3x = (3 * 2x + 2x * 2x) + -2
15 + 3x = (6x + 4x2) + -2

Reorder the terms:
15 + 3x = -2 + 6x + 4x2

Solving
15 + 3x = -2 + 6x + 4x2

Solving for variable 'x'.

Reorder the terms:
15 + 2 + 3x + -6x + -4x2 = -2 + 6x + 4x2 + 2 + -6x + -4x2

Combine like terms: 15 + 2 = 17
17 + 3x + -6x + -4x2 = -2 + 6x + 4x2 + 2 + -6x + -4x2

Combine like terms: 3x + -6x = -3x
17 + -3x + -4x2 = -2 + 6x + 4x2 + 2 + -6x + -4x2

Reorder the terms:
17 + -3x + -4x2 = -2 + 2 + 6x + -6x + 4x2 + -4x2

Combine like terms: -2 + 2 = 0
17 + -3x + -4x2 = 0 + 6x + -6x + 4x2 + -4x2
17 + -3x + -4x2 = 6x + -6x + 4x2 + -4x2

Combine like terms: 6x + -6x = 0
17 + -3x + -4x2 = 0 + 4x2 + -4x2
17 + -3x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
17 + -3x + -4x2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-4.25 + 0.75x + x2 = 0

Move the constant term to the right:

Add '4.25' to each side of the equation.
-4.25 + 0.75x + 4.25 + x2 = 0 + 4.25

Reorder the terms:
-4.25 + 4.25 + 0.75x + x2 = 0 + 4.25

Combine like terms: -4.25 + 4.25 = 0.00
0.00 + 0.75x + x2 = 0 + 4.25
0.75x + x2 = 0 + 4.25

Combine like terms: 0 + 4.25 = 4.25
0.75x + x2 = 4.25

The x term is 0.75x.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75x + 0.140625 + x2 = 4.25 + 0.140625

Reorder the terms:
0.140625 + 0.75x + x2 = 4.25 + 0.140625

Combine like terms: 4.25 + 0.140625 = 4.390625
0.140625 + 0.75x + x2 = 4.390625

Factor a perfect square on the left side:
(x + 0.375)(x + 0.375) = 4.390625

Calculate the square root of the right side: 2.095381827

Break this problem into two subproblems by setting 
(x + 0.375) equal to 2.095381827 and -2.095381827.

Subproblem 1

x + 0.375 = 2.095381827 Simplifying x + 0.375 = 2.095381827 Reorder the terms: 0.375 + x = 2.095381827 Solving 0.375 + x = 2.095381827 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + x = 2.095381827 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + x = 2.095381827 + -0.375 x = 2.095381827 + -0.375 Combine like terms: 2.095381827 + -0.375 = 1.720381827 x = 1.720381827 Simplifying x = 1.720381827

Subproblem 2

x + 0.375 = -2.095381827 Simplifying x + 0.375 = -2.095381827 Reorder the terms: 0.375 + x = -2.095381827 Solving 0.375 + x = -2.095381827 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + x = -2.095381827 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + x = -2.095381827 + -0.375 x = -2.095381827 + -0.375 Combine like terms: -2.095381827 + -0.375 = -2.470381827 x = -2.470381827 Simplifying x = -2.470381827

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.720381827, -2.470381827}

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