15x^2-67x-38=0

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Solution for 15x^2-67x-38=0 equation: 


Simplifying
15x2 + -67x + -38 = 0

Reorder the terms:
-38 + -67x + 15x2 = 0

Solving
-38 + -67x + 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'.
-2.533333333 + -4.466666667x + x2 = 0

Move the constant term to the right:

Add '2.533333333' to each side of the equation.
-2.533333333 + -4.466666667x + 2.533333333 + x2 = 0 + 2.533333333

Reorder the terms:
-2.533333333 + 2.533333333 + -4.466666667x + x2 = 0 + 2.533333333

Combine like terms: -2.533333333 + 2.533333333 = 0.000000000
0.000000000 + -4.466666667x + x2 = 0 + 2.533333333
-4.466666667x + x2 = 0 + 2.533333333

Combine like terms: 0 + 2.533333333 = 2.533333333
-4.466666667x + x2 = 2.533333333

The x term is -4.466666667x.  Take half its coefficient (-2.233333334).
Square it (4.987777781) and add it to both sides.

Add '4.987777781' to each side of the equation.
-4.466666667x + 4.987777781 + x2 = 2.533333333 + 4.987777781

Reorder the terms:
4.987777781 + -4.466666667x + x2 = 2.533333333 + 4.987777781

Combine like terms: 2.533333333 + 4.987777781 = 7.521111114
4.987777781 + -4.466666667x + x2 = 7.521111114

Factor a perfect square on the left side:
(x + -2.233333334)(x + -2.233333334) = 7.521111114

Calculate the square root of the right side: 2.742464423

Break this problem into two subproblems by setting 
(x + -2.233333334) equal to 2.742464423 and -2.742464423.


Subproblem 1
x + -2.233333334 = 2.742464423

Simplifying
x + -2.233333334 = 2.742464423

Reorder the terms:
-2.233333334 + x = 2.742464423

Solving
-2.233333334 + x = 2.742464423

Solving for variable 'x'.

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

Add '2.233333334' to each side of the equation.
-2.233333334 + 2.233333334 + x = 2.742464423 + 2.233333334

Combine like terms: -2.233333334 + 2.233333334 = 0.000000000
0.000000000 + x = 2.742464423 + 2.233333334
x = 2.742464423 + 2.233333334

Combine like terms: 2.742464423 + 2.233333334 = 4.975797757
x = 4.975797757

Simplifying
x = 4.975797757


Subproblem 2
x + -2.233333334 = -2.742464423

Simplifying
x + -2.233333334 = -2.742464423

Reorder the terms:
-2.233333334 + x = -2.742464423

Solving
-2.233333334 + x = -2.742464423

Solving for variable 'x'.

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

Add '2.233333334' to each side of the equation.
-2.233333334 + 2.233333334 + x = -2.742464423 + 2.233333334

Combine like terms: -2.233333334 + 2.233333334 = 0.000000000
0.000000000 + x = -2.742464423 + 2.233333334
x = -2.742464423 + 2.233333334

Combine like terms: -2.742464423 + 2.233333334 = -0.509131089
x = -0.509131089

Simplifying
x = -0.509131089


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.975797757, -0.509131089}

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