(4x+31)(6x+5)=180

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Solution for (4x+31)(6x+5)=180 equation:


Simplifying
(4x + 31)(6x + 5) = 180

Reorder the terms:
(31 + 4x)(6x + 5) = 180

Reorder the terms:
(31 + 4x)(5 + 6x) = 180

Multiply (31 + 4x) * (5 + 6x)
(31(5 + 6x) + 4x * (5 + 6x)) = 180
((5 * 31 + 6x * 31) + 4x * (5 + 6x)) = 180
((155 + 186x) + 4x * (5 + 6x)) = 180
(155 + 186x + (5 * 4x + 6x * 4x)) = 180
(155 + 186x + (20x + 24x2)) = 180

Combine like terms: 186x + 20x = 206x
(155 + 206x + 24x2) = 180

Solving
155 + 206x + 24x2 = 180

Solving for variable 'x'.

Reorder the terms:
155 + -180 + 206x + 24x2 = 180 + -180

Combine like terms: 155 + -180 = -25
-25 + 206x + 24x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-25 + 206x + 24x2 = 0

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

Divide each side by '24'.
-1.041666667 + 8.583333333x + x2 = 0

Move the constant term to the right:

Add '1.041666667' to each side of the equation.
-1.041666667 + 8.583333333x + 1.041666667 + x2 = 0 + 1.041666667

Reorder the terms:
-1.041666667 + 1.041666667 + 8.583333333x + x2 = 0 + 1.041666667

Combine like terms: -1.041666667 + 1.041666667 = 0.000000000
0.000000000 + 8.583333333x + x2 = 0 + 1.041666667
8.583333333x + x2 = 0 + 1.041666667

Combine like terms: 0 + 1.041666667 = 1.041666667
8.583333333x + x2 = 1.041666667

The x term is 8.583333333x.  Take half its coefficient (4.291666667).
Square it (18.41840278) and add it to both sides.

Add '18.41840278' to each side of the equation.
8.583333333x + 18.41840278 + x2 = 1.041666667 + 18.41840278

Reorder the terms:
18.41840278 + 8.583333333x + x2 = 1.041666667 + 18.41840278

Combine like terms: 1.041666667 + 18.41840278 = 19.460069447
18.41840278 + 8.583333333x + x2 = 19.460069447

Factor a perfect square on the left side:
(x + 4.291666667)(x + 4.291666667) = 19.460069447

Calculate the square root of the right side: 4.411356871

Break this problem into two subproblems by setting 
(x + 4.291666667) equal to 4.411356871 and -4.411356871.

Subproblem 1

x + 4.291666667 = 4.411356871 Simplifying x + 4.291666667 = 4.411356871 Reorder the terms: 4.291666667 + x = 4.411356871 Solving 4.291666667 + x = 4.411356871 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.291666667' to each side of the equation. 4.291666667 + -4.291666667 + x = 4.411356871 + -4.291666667 Combine like terms: 4.291666667 + -4.291666667 = 0.000000000 0.000000000 + x = 4.411356871 + -4.291666667 x = 4.411356871 + -4.291666667 Combine like terms: 4.411356871 + -4.291666667 = 0.119690204 x = 0.119690204 Simplifying x = 0.119690204

Subproblem 2

x + 4.291666667 = -4.411356871 Simplifying x + 4.291666667 = -4.411356871 Reorder the terms: 4.291666667 + x = -4.411356871 Solving 4.291666667 + x = -4.411356871 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.291666667' to each side of the equation. 4.291666667 + -4.291666667 + x = -4.411356871 + -4.291666667 Combine like terms: 4.291666667 + -4.291666667 = 0.000000000 0.000000000 + x = -4.411356871 + -4.291666667 x = -4.411356871 + -4.291666667 Combine like terms: -4.411356871 + -4.291666667 = -8.703023538 x = -8.703023538 Simplifying x = -8.703023538

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.119690204, -8.703023538}

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