(6x+15)(3x+14)+(9x+13)(6x+15)=0

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Solution for (6x+15)(3x+14)+(9x+13)(6x+15)=0 equation:


Simplifying
(6x + 15)(3x + 14) + (9x + 13)(6x + 15) = 0

Reorder the terms:
(15 + 6x)(3x + 14) + (9x + 13)(6x + 15) = 0

Reorder the terms:
(15 + 6x)(14 + 3x) + (9x + 13)(6x + 15) = 0

Multiply (15 + 6x) * (14 + 3x)
(15(14 + 3x) + 6x * (14 + 3x)) + (9x + 13)(6x + 15) = 0
((14 * 15 + 3x * 15) + 6x * (14 + 3x)) + (9x + 13)(6x + 15) = 0
((210 + 45x) + 6x * (14 + 3x)) + (9x + 13)(6x + 15) = 0
(210 + 45x + (14 * 6x + 3x * 6x)) + (9x + 13)(6x + 15) = 0
(210 + 45x + (84x + 18x2)) + (9x + 13)(6x + 15) = 0

Combine like terms: 45x + 84x = 129x
(210 + 129x + 18x2) + (9x + 13)(6x + 15) = 0

Reorder the terms:
210 + 129x + 18x2 + (13 + 9x)(6x + 15) = 0

Reorder the terms:
210 + 129x + 18x2 + (13 + 9x)(15 + 6x) = 0

Multiply (13 + 9x) * (15 + 6x)
210 + 129x + 18x2 + (13(15 + 6x) + 9x * (15 + 6x)) = 0
210 + 129x + 18x2 + ((15 * 13 + 6x * 13) + 9x * (15 + 6x)) = 0
210 + 129x + 18x2 + ((195 + 78x) + 9x * (15 + 6x)) = 0
210 + 129x + 18x2 + (195 + 78x + (15 * 9x + 6x * 9x)) = 0
210 + 129x + 18x2 + (195 + 78x + (135x + 54x2)) = 0

Combine like terms: 78x + 135x = 213x
210 + 129x + 18x2 + (195 + 213x + 54x2) = 0

Reorder the terms:
210 + 195 + 129x + 213x + 18x2 + 54x2 = 0

Combine like terms: 210 + 195 = 405
405 + 129x + 213x + 18x2 + 54x2 = 0

Combine like terms: 129x + 213x = 342x
405 + 342x + 18x2 + 54x2 = 0

Combine like terms: 18x2 + 54x2 = 72x2
405 + 342x + 72x2 = 0

Solving
405 + 342x + 72x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '9'.
9(45 + 38x + 8x2) = 0

Factor a trinomial.
9((5 + 2x)(9 + 4x)) = 0

Ignore the factor 9.

Subproblem 1

Set the factor '(5 + 2x)' equal to zero and attempt to solve: Simplifying 5 + 2x = 0 Solving 5 + 2x = 0 Move all terms containing x to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + 2x = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 2x = 0 + -5 2x = 0 + -5 Combine like terms: 0 + -5 = -5 2x = -5 Divide each side by '2'. x = -2.5 Simplifying x = -2.5

Subproblem 2

Set the factor '(9 + 4x)' equal to zero and attempt to solve: Simplifying 9 + 4x = 0 Solving 9 + 4x = 0 Move all terms containing x to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + 4x = 0 + -9 Combine like terms: 9 + -9 = 0 0 + 4x = 0 + -9 4x = 0 + -9 Combine like terms: 0 + -9 = -9 4x = -9 Divide each side by '4'. x = -2.25 Simplifying x = -2.25

Solution

x = {-2.5, -2.25}

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