15x^2-23x-41=0

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Solution for 15x^2-23x-41=0 equation:


Simplifying
15x2 + -23x + -41 = 0

Reorder the terms:
-41 + -23x + 15x2 = 0

Solving
-41 + -23x + 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'.
-2.733333333 + -1.533333333x + x2 = 0

Move the constant term to the right:

Add '2.733333333' to each side of the equation.
-2.733333333 + -1.533333333x + 2.733333333 + x2 = 0 + 2.733333333

Reorder the terms:
-2.733333333 + 2.733333333 + -1.533333333x + x2 = 0 + 2.733333333

Combine like terms: -2.733333333 + 2.733333333 = 0.000000000
0.000000000 + -1.533333333x + x2 = 0 + 2.733333333
-1.533333333x + x2 = 0 + 2.733333333

Combine like terms: 0 + 2.733333333 = 2.733333333
-1.533333333x + x2 = 2.733333333

The x term is -1.533333333x.  Take half its coefficient (-0.7666666665).
Square it (0.5877777775) and add it to both sides.

Add '0.5877777775' to each side of the equation.
-1.533333333x + 0.5877777775 + x2 = 2.733333333 + 0.5877777775

Reorder the terms:
0.5877777775 + -1.533333333x + x2 = 2.733333333 + 0.5877777775

Combine like terms: 2.733333333 + 0.5877777775 = 3.3211111105
0.5877777775 + -1.533333333x + x2 = 3.3211111105

Factor a perfect square on the left side:
(x + -0.7666666665)(x + -0.7666666665) = 3.3211111105

Calculate the square root of the right side: 1.822391591

Break this problem into two subproblems by setting 
(x + -0.7666666665) equal to 1.822391591 and -1.822391591.

Subproblem 1

x + -0.7666666665 = 1.822391591 Simplifying x + -0.7666666665 = 1.822391591 Reorder the terms: -0.7666666665 + x = 1.822391591 Solving -0.7666666665 + x = 1.822391591 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.7666666665' to each side of the equation. -0.7666666665 + 0.7666666665 + x = 1.822391591 + 0.7666666665 Combine like terms: -0.7666666665 + 0.7666666665 = 0.0000000000 0.0000000000 + x = 1.822391591 + 0.7666666665 x = 1.822391591 + 0.7666666665 Combine like terms: 1.822391591 + 0.7666666665 = 2.5890582575 x = 2.5890582575 Simplifying x = 2.5890582575

Subproblem 2

x + -0.7666666665 = -1.822391591 Simplifying x + -0.7666666665 = -1.822391591 Reorder the terms: -0.7666666665 + x = -1.822391591 Solving -0.7666666665 + x = -1.822391591 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.7666666665' to each side of the equation. -0.7666666665 + 0.7666666665 + x = -1.822391591 + 0.7666666665 Combine like terms: -0.7666666665 + 0.7666666665 = 0.0000000000 0.0000000000 + x = -1.822391591 + 0.7666666665 x = -1.822391591 + 0.7666666665 Combine like terms: -1.822391591 + 0.7666666665 = -1.0557249245 x = -1.0557249245 Simplifying x = -1.0557249245

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.5890582575, -1.0557249245}

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