24x^2-13x-13=0

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Solution for 24x^2-13x-13=0 equation: 


Simplifying
24x2 + -13x + -13 = 0

Reorder the terms:
-13 + -13x + 24x2 = 0

Solving
-13 + -13x + 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'.
-0.5416666667 + -0.5416666667x + x2 = 0

Move the constant term to the right:

Add '0.5416666667' to each side of the equation.
-0.5416666667 + -0.5416666667x + 0.5416666667 + x2 = 0 + 0.5416666667

Reorder the terms:
-0.5416666667 + 0.5416666667 + -0.5416666667x + x2 = 0 + 0.5416666667

Combine like terms: -0.5416666667 + 0.5416666667 = 0.0000000000
0.0000000000 + -0.5416666667x + x2 = 0 + 0.5416666667
-0.5416666667x + x2 = 0 + 0.5416666667

Combine like terms: 0 + 0.5416666667 = 0.5416666667
-0.5416666667x + x2 = 0.5416666667

The x term is -0.5416666667x.  Take half its coefficient (-0.2708333334).
Square it (0.07335069448) and add it to both sides.

Add '0.07335069448' to each side of the equation.
-0.5416666667x + 0.07335069448 + x2 = 0.5416666667 + 0.07335069448

Reorder the terms:
0.07335069448 + -0.5416666667x + x2 = 0.5416666667 + 0.07335069448

Combine like terms: 0.5416666667 + 0.07335069448 = 0.61501736118
0.07335069448 + -0.5416666667x + x2 = 0.61501736118

Factor a perfect square on the left side:
(x + -0.2708333334)(x + -0.2708333334) = 0.61501736118

Calculate the square root of the right side: 0.784230426

Break this problem into two subproblems by setting 
(x + -0.2708333334) equal to 0.784230426 and -0.784230426.


Subproblem 1
x + -0.2708333334 = 0.784230426

Simplifying
x + -0.2708333334 = 0.784230426

Reorder the terms:
-0.2708333334 + x = 0.784230426

Solving
-0.2708333334 + x = 0.784230426

Solving for variable 'x'.

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

Add '0.2708333334' to each side of the equation.
-0.2708333334 + 0.2708333334 + x = 0.784230426 + 0.2708333334

Combine like terms: -0.2708333334 + 0.2708333334 = 0.0000000000
0.0000000000 + x = 0.784230426 + 0.2708333334
x = 0.784230426 + 0.2708333334

Combine like terms: 0.784230426 + 0.2708333334 = 1.0550637594
x = 1.0550637594

Simplifying
x = 1.0550637594


Subproblem 2
x + -0.2708333334 = -0.784230426

Simplifying
x + -0.2708333334 = -0.784230426

Reorder the terms:
-0.2708333334 + x = -0.784230426

Solving
-0.2708333334 + x = -0.784230426

Solving for variable 'x'.

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

Add '0.2708333334' to each side of the equation.
-0.2708333334 + 0.2708333334 + x = -0.784230426 + 0.2708333334

Combine like terms: -0.2708333334 + 0.2708333334 = 0.0000000000
0.0000000000 + x = -0.784230426 + 0.2708333334
x = -0.784230426 + 0.2708333334

Combine like terms: -0.784230426 + 0.2708333334 = -0.5133970926
x = -0.5133970926

Simplifying
x = -0.5133970926


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.0550637594, -0.5133970926}

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