(6x+7)(5x-3)=180

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Solution for (6x+7)(5x-3)=180 equation:


Simplifying
(6x + 7)(5x + -3) = 180

Reorder the terms:
(7 + 6x)(5x + -3) = 180

Reorder the terms:
(7 + 6x)(-3 + 5x) = 180

Multiply (7 + 6x) * (-3 + 5x)
(7(-3 + 5x) + 6x * (-3 + 5x)) = 180
((-3 * 7 + 5x * 7) + 6x * (-3 + 5x)) = 180
((-21 + 35x) + 6x * (-3 + 5x)) = 180
(-21 + 35x + (-3 * 6x + 5x * 6x)) = 180
(-21 + 35x + (-18x + 30x2)) = 180

Combine like terms: 35x + -18x = 17x
(-21 + 17x + 30x2) = 180

Solving
-21 + 17x + 30x2 = 180

Solving for variable 'x'.

Reorder the terms:
-21 + -180 + 17x + 30x2 = 180 + -180

Combine like terms: -21 + -180 = -201
-201 + 17x + 30x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-201 + 17x + 30x2 = 0

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

Divide each side by '30'.
-6.7 + 0.5666666667x + x2 = 0.0

Move the constant term to the right:

Add '6.7' to each side of the equation.
-6.7 + 0.5666666667x + 6.7 + x2 = 0.0 + 6.7

Reorder the terms:
-6.7 + 6.7 + 0.5666666667x + x2 = 0.0 + 6.7

Combine like terms: -6.7 + 6.7 = 0.0
0.0 + 0.5666666667x + x2 = 0.0 + 6.7
0.5666666667x + x2 = 0.0 + 6.7

Combine like terms: 0.0 + 6.7 = 6.7
0.5666666667x + x2 = 6.7

The x term is 0.5666666667x.  Take half its coefficient (0.2833333334).
Square it (0.08027777782) and add it to both sides.

Add '0.08027777782' to each side of the equation.
0.5666666667x + 0.08027777782 + x2 = 6.7 + 0.08027777782

Reorder the terms:
0.08027777782 + 0.5666666667x + x2 = 6.7 + 0.08027777782

Combine like terms: 6.7 + 0.08027777782 = 6.78027777782
0.08027777782 + 0.5666666667x + x2 = 6.78027777782

Factor a perfect square on the left side:
(x + 0.2833333334)(x + 0.2833333334) = 6.78027777782

Calculate the square root of the right side: 2.603896653

Break this problem into two subproblems by setting 
(x + 0.2833333334) equal to 2.603896653 and -2.603896653.

Subproblem 1

x + 0.2833333334 = 2.603896653 Simplifying x + 0.2833333334 = 2.603896653 Reorder the terms: 0.2833333334 + x = 2.603896653 Solving 0.2833333334 + x = 2.603896653 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2833333334' to each side of the equation. 0.2833333334 + -0.2833333334 + x = 2.603896653 + -0.2833333334 Combine like terms: 0.2833333334 + -0.2833333334 = 0.0000000000 0.0000000000 + x = 2.603896653 + -0.2833333334 x = 2.603896653 + -0.2833333334 Combine like terms: 2.603896653 + -0.2833333334 = 2.3205633196 x = 2.3205633196 Simplifying x = 2.3205633196

Subproblem 2

x + 0.2833333334 = -2.603896653 Simplifying x + 0.2833333334 = -2.603896653 Reorder the terms: 0.2833333334 + x = -2.603896653 Solving 0.2833333334 + x = -2.603896653 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2833333334' to each side of the equation. 0.2833333334 + -0.2833333334 + x = -2.603896653 + -0.2833333334 Combine like terms: 0.2833333334 + -0.2833333334 = 0.0000000000 0.0000000000 + x = -2.603896653 + -0.2833333334 x = -2.603896653 + -0.2833333334 Combine like terms: -2.603896653 + -0.2833333334 = -2.8872299864 x = -2.8872299864 Simplifying x = -2.8872299864

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.3205633196, -2.8872299864}

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