24x^2-38x-15=0

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Solution for 24x^2-38x-15=0 equation:


Simplifying
24x2 + -38x + -15 = 0

Reorder the terms:
-15 + -38x + 24x2 = 0

Solving
-15 + -38x + 24x2 = 0

Solving for variable 'x'.

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

Divide each side by '24'.
-0.625 + -1.583333333x + x2 = 0

Move the constant term to the right:

Add '0.625' to each side of the equation.
-0.625 + -1.583333333x + 0.625 + x2 = 0 + 0.625

Reorder the terms:
-0.625 + 0.625 + -1.583333333x + x2 = 0 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + -1.583333333x + x2 = 0 + 0.625
-1.583333333x + x2 = 0 + 0.625

Combine like terms: 0 + 0.625 = 0.625
-1.583333333x + x2 = 0.625

The x term is -1.583333333x.  Take half its coefficient (-0.7916666665).
Square it (0.6267361108) and add it to both sides.

Add '0.6267361108' to each side of the equation.
-1.583333333x + 0.6267361108 + x2 = 0.625 + 0.6267361108

Reorder the terms:
0.6267361108 + -1.583333333x + x2 = 0.625 + 0.6267361108

Combine like terms: 0.625 + 0.6267361108 = 1.2517361108
0.6267361108 + -1.583333333x + x2 = 1.2517361108

Factor a perfect square on the left side:
(x + -0.7916666665)(x + -0.7916666665) = 1.2517361108

Calculate the square root of the right side: 1.118810132

Break this problem into two subproblems by setting 
(x + -0.7916666665) equal to 1.118810132 and -1.118810132.

Subproblem 1

x + -0.7916666665 = 1.118810132 Simplifying x + -0.7916666665 = 1.118810132 Reorder the terms: -0.7916666665 + x = 1.118810132 Solving -0.7916666665 + x = 1.118810132 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.7916666665' to each side of the equation. -0.7916666665 + 0.7916666665 + x = 1.118810132 + 0.7916666665 Combine like terms: -0.7916666665 + 0.7916666665 = 0.0000000000 0.0000000000 + x = 1.118810132 + 0.7916666665 x = 1.118810132 + 0.7916666665 Combine like terms: 1.118810132 + 0.7916666665 = 1.9104767985 x = 1.9104767985 Simplifying x = 1.9104767985

Subproblem 2

x + -0.7916666665 = -1.118810132 Simplifying x + -0.7916666665 = -1.118810132 Reorder the terms: -0.7916666665 + x = -1.118810132 Solving -0.7916666665 + x = -1.118810132 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.7916666665' to each side of the equation. -0.7916666665 + 0.7916666665 + x = -1.118810132 + 0.7916666665 Combine like terms: -0.7916666665 + 0.7916666665 = 0.0000000000 0.0000000000 + x = -1.118810132 + 0.7916666665 x = -1.118810132 + 0.7916666665 Combine like terms: -1.118810132 + 0.7916666665 = -0.3271434655 x = -0.3271434655 Simplifying x = -0.3271434655

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.9104767985, -0.3271434655}

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