15x^2+85x+60=0

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Solution for 15x^2+85x+60=0 equation: 


Simplifying
15x2 + 85x + 60 = 0

Reorder the terms:
60 + 85x + 15x2 = 0

Solving
60 + 85x + 15x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '5'.
5(12 + 17x + 3x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(12 + 17x + 3x2)' equal to zero and attempt to solve:

Simplifying
12 + 17x + 3x2 = 0

Solving
12 + 17x + 3x2 = 0

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

Divide each side by '3'.
4 + 5.666666667x + x2 = 0

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + 5.666666667x + -4 + x2 = 0 + -4

Reorder the terms:
4 + -4 + 5.666666667x + x2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + 5.666666667x + x2 = 0 + -4
5.666666667x + x2 = 0 + -4

Combine like terms: 0 + -4 = -4
5.666666667x + x2 = -4

The x term is 5.666666667x.  Take half its coefficient (2.833333334).
Square it (8.027777782) and add it to both sides.

Add '8.027777782' to each side of the equation.
5.666666667x + 8.027777782 + x2 = -4 + 8.027777782

Reorder the terms:
8.027777782 + 5.666666667x + x2 = -4 + 8.027777782

Combine like terms: -4 + 8.027777782 = 4.027777782
8.027777782 + 5.666666667x + x2 = 4.027777782

Factor a perfect square on the left side:
(x + 2.833333334)(x + 2.833333334) = 4.027777782

Calculate the square root of the right side: 2.006932431

Break this problem into two subproblems by setting 
(x + 2.833333334) equal to 2.006932431 and -2.006932431.


Subproblem 1
x + 2.833333334 = 2.006932431

Simplifying
x + 2.833333334 = 2.006932431

Reorder the terms:
2.833333334 + x = 2.006932431

Solving
2.833333334 + x = 2.006932431

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-2.833333334' to each side of the equation.
2.833333334 + -2.833333334 + x = 2.006932431 + -2.833333334

Combine like terms: 2.833333334 + -2.833333334 = 0.000000000
0.000000000 + x = 2.006932431 + -2.833333334
x = 2.006932431 + -2.833333334

Combine like terms: 2.006932431 + -2.833333334 = -0.826400903
x = -0.826400903

Simplifying
x = -0.826400903


Subproblem 2
x + 2.833333334 = -2.006932431

Simplifying
x + 2.833333334 = -2.006932431

Reorder the terms:
2.833333334 + x = -2.006932431

Solving
2.833333334 + x = -2.006932431

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-2.833333334' to each side of the equation.
2.833333334 + -2.833333334 + x = -2.006932431 + -2.833333334

Combine like terms: 2.833333334 + -2.833333334 = 0.000000000
0.000000000 + x = -2.006932431 + -2.833333334
x = -2.006932431 + -2.833333334

Combine like terms: -2.006932431 + -2.833333334 = -4.840265765
x = -4.840265765

Simplifying
x = -4.840265765


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.826400903, -4.840265765}
Solution
x = {-0.826400903, -4.840265765}

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