(2x+21)(3x-4)=180

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Solution for (2x+21)(3x-4)=180 equation:


Simplifying
(2x + 21)(3x + -4) = 180

Reorder the terms:
(21 + 2x)(3x + -4) = 180

Reorder the terms:
(21 + 2x)(-4 + 3x) = 180

Multiply (21 + 2x) * (-4 + 3x)
(21(-4 + 3x) + 2x * (-4 + 3x)) = 180
((-4 * 21 + 3x * 21) + 2x * (-4 + 3x)) = 180
((-84 + 63x) + 2x * (-4 + 3x)) = 180
(-84 + 63x + (-4 * 2x + 3x * 2x)) = 180
(-84 + 63x + (-8x + 6x2)) = 180

Combine like terms: 63x + -8x = 55x
(-84 + 55x + 6x2) = 180

Solving
-84 + 55x + 6x2 = 180

Solving for variable 'x'.

Reorder the terms:
-84 + -180 + 55x + 6x2 = 180 + -180

Combine like terms: -84 + -180 = -264
-264 + 55x + 6x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-264 + 55x + 6x2 = 0

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

Divide each side by '6'.
-44 + 9.166666667x + x2 = 0

Move the constant term to the right:

Add '44' to each side of the equation.
-44 + 9.166666667x + 44 + x2 = 0 + 44

Reorder the terms:
-44 + 44 + 9.166666667x + x2 = 0 + 44

Combine like terms: -44 + 44 = 0
0 + 9.166666667x + x2 = 0 + 44
9.166666667x + x2 = 0 + 44

Combine like terms: 0 + 44 = 44
9.166666667x + x2 = 44

The x term is 9.166666667x.  Take half its coefficient (4.583333334).
Square it (21.00694445) and add it to both sides.

Add '21.00694445' to each side of the equation.
9.166666667x + 21.00694445 + x2 = 44 + 21.00694445

Reorder the terms:
21.00694445 + 9.166666667x + x2 = 44 + 21.00694445

Combine like terms: 44 + 21.00694445 = 65.00694445
21.00694445 + 9.166666667x + x2 = 65.00694445

Factor a perfect square on the left side:
(x + 4.583333334)(x + 4.583333334) = 65.00694445

Calculate the square root of the right side: 8.062688413

Break this problem into two subproblems by setting 
(x + 4.583333334) equal to 8.062688413 and -8.062688413.

Subproblem 1

x + 4.583333334 = 8.062688413 Simplifying x + 4.583333334 = 8.062688413 Reorder the terms: 4.583333334 + x = 8.062688413 Solving 4.583333334 + x = 8.062688413 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.583333334' to each side of the equation. 4.583333334 + -4.583333334 + x = 8.062688413 + -4.583333334 Combine like terms: 4.583333334 + -4.583333334 = 0.000000000 0.000000000 + x = 8.062688413 + -4.583333334 x = 8.062688413 + -4.583333334 Combine like terms: 8.062688413 + -4.583333334 = 3.479355079 x = 3.479355079 Simplifying x = 3.479355079

Subproblem 2

x + 4.583333334 = -8.062688413 Simplifying x + 4.583333334 = -8.062688413 Reorder the terms: 4.583333334 + x = -8.062688413 Solving 4.583333334 + x = -8.062688413 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.583333334' to each side of the equation. 4.583333334 + -4.583333334 + x = -8.062688413 + -4.583333334 Combine like terms: 4.583333334 + -4.583333334 = 0.000000000 0.000000000 + x = -8.062688413 + -4.583333334 x = -8.062688413 + -4.583333334 Combine like terms: -8.062688413 + -4.583333334 = -12.646021747 x = -12.646021747 Simplifying x = -12.646021747

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.479355079, -12.646021747}

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