(3x+13)(3x-5)=-24

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Solution for (3x+13)(3x-5)=-24 equation:


Simplifying
(3x + 13)(3x + -5) = -24

Reorder the terms:
(13 + 3x)(3x + -5) = -24

Reorder the terms:
(13 + 3x)(-5 + 3x) = -24

Multiply (13 + 3x) * (-5 + 3x)
(13(-5 + 3x) + 3x * (-5 + 3x)) = -24
((-5 * 13 + 3x * 13) + 3x * (-5 + 3x)) = -24
((-65 + 39x) + 3x * (-5 + 3x)) = -24
(-65 + 39x + (-5 * 3x + 3x * 3x)) = -24
(-65 + 39x + (-15x + 9x2)) = -24

Combine like terms: 39x + -15x = 24x
(-65 + 24x + 9x2) = -24

Solving
-65 + 24x + 9x2 = -24

Solving for variable 'x'.

Reorder the terms:
-65 + 24 + 24x + 9x2 = -24 + 24

Combine like terms: -65 + 24 = -41
-41 + 24x + 9x2 = -24 + 24

Combine like terms: -24 + 24 = 0
-41 + 24x + 9x2 = 0

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

Divide each side by '9'.
-4.555555556 + 2.666666667x + x2 = 0

Move the constant term to the right:

Add '4.555555556' to each side of the equation.
-4.555555556 + 2.666666667x + 4.555555556 + x2 = 0 + 4.555555556

Reorder the terms:
-4.555555556 + 4.555555556 + 2.666666667x + x2 = 0 + 4.555555556

Combine like terms: -4.555555556 + 4.555555556 = 0.000000000
0.000000000 + 2.666666667x + x2 = 0 + 4.555555556
2.666666667x + x2 = 0 + 4.555555556

Combine like terms: 0 + 4.555555556 = 4.555555556
2.666666667x + x2 = 4.555555556

The x term is 2.666666667x.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667x + 1.777777780 + x2 = 4.555555556 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667x + x2 = 4.555555556 + 1.777777780

Combine like terms: 4.555555556 + 1.777777780 = 6.333333336
1.777777780 + 2.666666667x + x2 = 6.333333336

Factor a perfect square on the left side:
(x + 1.333333334)(x + 1.333333334) = 6.333333336

Calculate the square root of the right side: 2.516611479

Break this problem into two subproblems by setting 
(x + 1.333333334) equal to 2.516611479 and -2.516611479.

Subproblem 1

x + 1.333333334 = 2.516611479 Simplifying x + 1.333333334 = 2.516611479 Reorder the terms: 1.333333334 + x = 2.516611479 Solving 1.333333334 + x = 2.516611479 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = 2.516611479 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 2.516611479 + -1.333333334 x = 2.516611479 + -1.333333334 Combine like terms: 2.516611479 + -1.333333334 = 1.183278145 x = 1.183278145 Simplifying x = 1.183278145

Subproblem 2

x + 1.333333334 = -2.516611479 Simplifying x + 1.333333334 = -2.516611479 Reorder the terms: 1.333333334 + x = -2.516611479 Solving 1.333333334 + x = -2.516611479 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = -2.516611479 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -2.516611479 + -1.333333334 x = -2.516611479 + -1.333333334 Combine like terms: -2.516611479 + -1.333333334 = -3.849944813 x = -3.849944813 Simplifying x = -3.849944813

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.183278145, -3.849944813}

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