(3x-7)(4x+5)=180

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Solution for (3x-7)(4x+5)=180 equation:


Simplifying
(3x + -7)(4x + 5) = 180

Reorder the terms:
(-7 + 3x)(4x + 5) = 180

Reorder the terms:
(-7 + 3x)(5 + 4x) = 180

Multiply (-7 + 3x) * (5 + 4x)
(-7(5 + 4x) + 3x * (5 + 4x)) = 180
((5 * -7 + 4x * -7) + 3x * (5 + 4x)) = 180
((-35 + -28x) + 3x * (5 + 4x)) = 180
(-35 + -28x + (5 * 3x + 4x * 3x)) = 180
(-35 + -28x + (15x + 12x2)) = 180

Combine like terms: -28x + 15x = -13x
(-35 + -13x + 12x2) = 180

Solving
-35 + -13x + 12x2 = 180

Solving for variable 'x'.

Reorder the terms:
-35 + -180 + -13x + 12x2 = 180 + -180

Combine like terms: -35 + -180 = -215
-215 + -13x + 12x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-215 + -13x + 12x2 = 0

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

Divide each side by '12'.
-17.91666667 + -1.083333333x + x2 = 0

Move the constant term to the right:

Add '17.91666667' to each side of the equation.
-17.91666667 + -1.083333333x + 17.91666667 + x2 = 0 + 17.91666667

Reorder the terms:
-17.91666667 + 17.91666667 + -1.083333333x + x2 = 0 + 17.91666667

Combine like terms: -17.91666667 + 17.91666667 = 0.00000000
0.00000000 + -1.083333333x + x2 = 0 + 17.91666667
-1.083333333x + x2 = 0 + 17.91666667

Combine like terms: 0 + 17.91666667 = 17.91666667
-1.083333333x + x2 = 17.91666667

The x term is -1.083333333x.  Take half its coefficient (-0.5416666665).
Square it (0.2934027776) and add it to both sides.

Add '0.2934027776' to each side of the equation.
-1.083333333x + 0.2934027776 + x2 = 17.91666667 + 0.2934027776

Reorder the terms:
0.2934027776 + -1.083333333x + x2 = 17.91666667 + 0.2934027776

Combine like terms: 17.91666667 + 0.2934027776 = 18.2100694476
0.2934027776 + -1.083333333x + x2 = 18.2100694476

Factor a perfect square on the left side:
(x + -0.5416666665)(x + -0.5416666665) = 18.2100694476

Calculate the square root of the right side: 4.267325796

Break this problem into two subproblems by setting 
(x + -0.5416666665) equal to 4.267325796 and -4.267325796.

Subproblem 1

x + -0.5416666665 = 4.267325796 Simplifying x + -0.5416666665 = 4.267325796 Reorder the terms: -0.5416666665 + x = 4.267325796 Solving -0.5416666665 + x = 4.267325796 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5416666665' to each side of the equation. -0.5416666665 + 0.5416666665 + x = 4.267325796 + 0.5416666665 Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000 0.0000000000 + x = 4.267325796 + 0.5416666665 x = 4.267325796 + 0.5416666665 Combine like terms: 4.267325796 + 0.5416666665 = 4.8089924625 x = 4.8089924625 Simplifying x = 4.8089924625

Subproblem 2

x + -0.5416666665 = -4.267325796 Simplifying x + -0.5416666665 = -4.267325796 Reorder the terms: -0.5416666665 + x = -4.267325796 Solving -0.5416666665 + x = -4.267325796 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5416666665' to each side of the equation. -0.5416666665 + 0.5416666665 + x = -4.267325796 + 0.5416666665 Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000 0.0000000000 + x = -4.267325796 + 0.5416666665 x = -4.267325796 + 0.5416666665 Combine like terms: -4.267325796 + 0.5416666665 = -3.7256591295 x = -3.7256591295 Simplifying x = -3.7256591295

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.8089924625, -3.7256591295}

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