3x-5=2x(7-x)+2

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Solution for 3x-5=2x(7-x)+2 equation:


Simplifying
3x + -5 = 2x(7 + -1x) + 2

Reorder the terms:
-5 + 3x = 2x(7 + -1x) + 2
-5 + 3x = (7 * 2x + -1x * 2x) + 2
-5 + 3x = (14x + -2x2) + 2

Reorder the terms:
-5 + 3x = 2 + 14x + -2x2

Solving
-5 + 3x = 2 + 14x + -2x2

Solving for variable 'x'.

Reorder the terms:
-5 + -2 + 3x + -14x + 2x2 = 2 + 14x + -2x2 + -2 + -14x + 2x2

Combine like terms: -5 + -2 = -7
-7 + 3x + -14x + 2x2 = 2 + 14x + -2x2 + -2 + -14x + 2x2

Combine like terms: 3x + -14x = -11x
-7 + -11x + 2x2 = 2 + 14x + -2x2 + -2 + -14x + 2x2

Reorder the terms:
-7 + -11x + 2x2 = 2 + -2 + 14x + -14x + -2x2 + 2x2

Combine like terms: 2 + -2 = 0
-7 + -11x + 2x2 = 0 + 14x + -14x + -2x2 + 2x2
-7 + -11x + 2x2 = 14x + -14x + -2x2 + 2x2

Combine like terms: 14x + -14x = 0
-7 + -11x + 2x2 = 0 + -2x2 + 2x2
-7 + -11x + 2x2 = -2x2 + 2x2

Combine like terms: -2x2 + 2x2 = 0
-7 + -11x + 2x2 = 0

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

Divide each side by '2'.
-3.5 + -5.5x + x2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + -5.5x + 3.5 + x2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + -5.5x + x2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + -5.5x + x2 = 0 + 3.5
-5.5x + x2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
-5.5x + x2 = 3.5

The x term is -5.5x.  Take half its coefficient (-2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
-5.5x + 7.5625 + x2 = 3.5 + 7.5625

Reorder the terms:
7.5625 + -5.5x + x2 = 3.5 + 7.5625

Combine like terms: 3.5 + 7.5625 = 11.0625
7.5625 + -5.5x + x2 = 11.0625

Factor a perfect square on the left side:
(x + -2.75)(x + -2.75) = 11.0625

Calculate the square root of the right side: 3.326033674

Break this problem into two subproblems by setting 
(x + -2.75) equal to 3.326033674 and -3.326033674.

Subproblem 1

x + -2.75 = 3.326033674 Simplifying x + -2.75 = 3.326033674 Reorder the terms: -2.75 + x = 3.326033674 Solving -2.75 + x = 3.326033674 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + x = 3.326033674 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + x = 3.326033674 + 2.75 x = 3.326033674 + 2.75 Combine like terms: 3.326033674 + 2.75 = 6.076033674 x = 6.076033674 Simplifying x = 6.076033674

Subproblem 2

x + -2.75 = -3.326033674 Simplifying x + -2.75 = -3.326033674 Reorder the terms: -2.75 + x = -3.326033674 Solving -2.75 + x = -3.326033674 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + x = -3.326033674 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + x = -3.326033674 + 2.75 x = -3.326033674 + 2.75 Combine like terms: -3.326033674 + 2.75 = -0.576033674 x = -0.576033674 Simplifying x = -0.576033674

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.076033674, -0.576033674}

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