15x^2+19x+5=0

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Solution for 15x^2+19x+5=0 equation: 


Simplifying
15x2 + 19x + 5 = 0

Reorder the terms:
5 + 19x + 15x2 = 0

Solving
5 + 19x + 15x2 = 0

Solving for variable 'x'.

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

Divide each side by '15'.
0.3333333333 + 1.266666667x + x2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + 1.266666667x + -0.3333333333 + x2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + 1.266666667x + x2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + 1.266666667x + x2 = 0 + -0.3333333333
1.266666667x + x2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
1.266666667x + x2 = -0.3333333333

The x term is 1.266666667x.  Take half its coefficient (0.6333333335).
Square it (0.4011111113) and add it to both sides.

Add '0.4011111113' to each side of the equation.
1.266666667x + 0.4011111113 + x2 = -0.3333333333 + 0.4011111113

Reorder the terms:
0.4011111113 + 1.266666667x + x2 = -0.3333333333 + 0.4011111113

Combine like terms: -0.3333333333 + 0.4011111113 = 0.067777778
0.4011111113 + 1.266666667x + x2 = 0.067777778

Factor a perfect square on the left side:
(x + 0.6333333335)(x + 0.6333333335) = 0.067777778

Calculate the square root of the right side: 0.260341656

Break this problem into two subproblems by setting 
(x + 0.6333333335) equal to 0.260341656 and -0.260341656.


Subproblem 1
x + 0.6333333335 = 0.260341656

Simplifying
x + 0.6333333335 = 0.260341656

Reorder the terms:
0.6333333335 + x = 0.260341656

Solving
0.6333333335 + x = 0.260341656

Solving for variable 'x'.

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

Add '-0.6333333335' to each side of the equation.
0.6333333335 + -0.6333333335 + x = 0.260341656 + -0.6333333335

Combine like terms: 0.6333333335 + -0.6333333335 = 0.0000000000
0.0000000000 + x = 0.260341656 + -0.6333333335
x = 0.260341656 + -0.6333333335

Combine like terms: 0.260341656 + -0.6333333335 = -0.3729916775
x = -0.3729916775

Simplifying
x = -0.3729916775


Subproblem 2
x + 0.6333333335 = -0.260341656

Simplifying
x + 0.6333333335 = -0.260341656

Reorder the terms:
0.6333333335 + x = -0.260341656

Solving
0.6333333335 + x = -0.260341656

Solving for variable 'x'.

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

Add '-0.6333333335' to each side of the equation.
0.6333333335 + -0.6333333335 + x = -0.260341656 + -0.6333333335

Combine like terms: 0.6333333335 + -0.6333333335 = 0.0000000000
0.0000000000 + x = -0.260341656 + -0.6333333335
x = -0.260341656 + -0.6333333335

Combine like terms: -0.260341656 + -0.6333333335 = -0.8936749895
x = -0.8936749895

Simplifying
x = -0.8936749895


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.3729916775, -0.8936749895}

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