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 = 0.7344444446
0.4011111113 + -1.266666667x + x2 = 0.7344444446

Factor a perfect square on the left side:
(x + -0.6333333335)(x + -0.6333333335) = 0.7344444446

Calculate the square root of the right side: 0.856997342

Break this problem into two subproblems by setting 
(x + -0.6333333335) equal to 0.856997342 and -0.856997342.


Subproblem 1
x + -0.6333333335 = 0.856997342

Simplifying
x + -0.6333333335 = 0.856997342

Reorder the terms:
-0.6333333335 + x = 0.856997342

Solving
-0.6333333335 + x = 0.856997342

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 = 0.856997342 + 0.6333333335

Combine like terms: -0.6333333335 + 0.6333333335 = 0.0000000000
0.0000000000 + x = 0.856997342 + 0.6333333335
x = 0.856997342 + 0.6333333335

Combine like terms: 0.856997342 + 0.6333333335 = 1.4903306755
x = 1.4903306755

Simplifying
x = 1.4903306755


Subproblem 2
x + -0.6333333335 = -0.856997342

Simplifying
x + -0.6333333335 = -0.856997342

Reorder the terms:
-0.6333333335 + x = -0.856997342

Solving
-0.6333333335 + x = -0.856997342

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 = -0.856997342 + 0.6333333335

Combine like terms: -0.6333333335 + 0.6333333335 = 0.0000000000
0.0000000000 + x = -0.856997342 + 0.6333333335
x = -0.856997342 + 0.6333333335

Combine like terms: -0.856997342 + 0.6333333335 = -0.2236640085
x = -0.2236640085

Simplifying
x = -0.2236640085


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.4903306755, -0.2236640085}

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