6x^2-29x-75=0

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Solution for 6x^2-29x-75=0 equation: 


Simplifying
6x2 + -29x + -75 = 0

Reorder the terms:
-75 + -29x + 6x2 = 0

Solving
-75 + -29x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-12.5 + -4.833333333x + x2 = 0

Move the constant term to the right:

Add '12.5' to each side of the equation.
-12.5 + -4.833333333x + 12.5 + x2 = 0 + 12.5

Reorder the terms:
-12.5 + 12.5 + -4.833333333x + x2 = 0 + 12.5

Combine like terms: -12.5 + 12.5 = 0.0
0.0 + -4.833333333x + x2 = 0 + 12.5
-4.833333333x + x2 = 0 + 12.5

Combine like terms: 0 + 12.5 = 12.5
-4.833333333x + x2 = 12.5

The x term is -4.833333333x.  Take half its coefficient (-2.416666667).
Square it (5.840277779) and add it to both sides.

Add '5.840277779' to each side of the equation.
-4.833333333x + 5.840277779 + x2 = 12.5 + 5.840277779

Reorder the terms:
5.840277779 + -4.833333333x + x2 = 12.5 + 5.840277779

Combine like terms: 12.5 + 5.840277779 = 18.340277779
5.840277779 + -4.833333333x + x2 = 18.340277779

Factor a perfect square on the left side:
(x + -2.416666667)(x + -2.416666667) = 18.340277779

Calculate the square root of the right side: 4.282555053

Break this problem into two subproblems by setting 
(x + -2.416666667) equal to 4.282555053 and -4.282555053.


Subproblem 1
x + -2.416666667 = 4.282555053

Simplifying
x + -2.416666667 = 4.282555053

Reorder the terms:
-2.416666667 + x = 4.282555053

Solving
-2.416666667 + x = 4.282555053

Solving for variable 'x'.

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

Add '2.416666667' to each side of the equation.
-2.416666667 + 2.416666667 + x = 4.282555053 + 2.416666667

Combine like terms: -2.416666667 + 2.416666667 = 0.000000000
0.000000000 + x = 4.282555053 + 2.416666667
x = 4.282555053 + 2.416666667

Combine like terms: 4.282555053 + 2.416666667 = 6.69922172
x = 6.69922172

Simplifying
x = 6.69922172


Subproblem 2
x + -2.416666667 = -4.282555053

Simplifying
x + -2.416666667 = -4.282555053

Reorder the terms:
-2.416666667 + x = -4.282555053

Solving
-2.416666667 + x = -4.282555053

Solving for variable 'x'.

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

Add '2.416666667' to each side of the equation.
-2.416666667 + 2.416666667 + x = -4.282555053 + 2.416666667

Combine like terms: -2.416666667 + 2.416666667 = 0.000000000
0.000000000 + x = -4.282555053 + 2.416666667
x = -4.282555053 + 2.416666667

Combine like terms: -4.282555053 + 2.416666667 = -1.865888386
x = -1.865888386

Simplifying
x = -1.865888386


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.69922172, -1.865888386}

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