3x^2-2x-75=0

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Solution for 3x^2-2x-75=0 equation: 


Simplifying
3x2 + -2x + -75 = 0

Reorder the terms:
-75 + -2x + 3x2 = 0

Solving
-75 + -2x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
-25 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + -0.6666666667x + 25 + x2 = 0 + 25

Reorder the terms:
-25 + 25 + -0.6666666667x + x2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + -0.6666666667x + x2 = 0 + 25
-0.6666666667x + x2 = 0 + 25

Combine like terms: 0 + 25 = 25
-0.6666666667x + x2 = 25

The x term is -0.6666666667x.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667x + 0.1111111112 + x2 = 25 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 25 + 0.1111111112

Combine like terms: 25 + 0.1111111112 = 25.1111111112
0.1111111112 + -0.6666666667x + x2 = 25.1111111112

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 25.1111111112

Calculate the square root of the right side: 5.011098793

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 5.011098793 and -5.011098793.


Subproblem 1
x + -0.3333333334 = 5.011098793

Simplifying
x + -0.3333333334 = 5.011098793

Reorder the terms:
-0.3333333334 + x = 5.011098793

Solving
-0.3333333334 + x = 5.011098793

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = 5.011098793 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = 5.011098793 + 0.3333333334
x = 5.011098793 + 0.3333333334

Combine like terms: 5.011098793 + 0.3333333334 = 5.3444321264
x = 5.3444321264

Simplifying
x = 5.3444321264


Subproblem 2
x + -0.3333333334 = -5.011098793

Simplifying
x + -0.3333333334 = -5.011098793

Reorder the terms:
-0.3333333334 + x = -5.011098793

Solving
-0.3333333334 + x = -5.011098793

Solving for variable 'x'.

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

Add '0.3333333334' to each side of the equation.
-0.3333333334 + 0.3333333334 + x = -5.011098793 + 0.3333333334

Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000
0.0000000000 + x = -5.011098793 + 0.3333333334
x = -5.011098793 + 0.3333333334

Combine like terms: -5.011098793 + 0.3333333334 = -4.6777654596
x = -4.6777654596

Simplifying
x = -4.6777654596


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.3444321264, -4.6777654596}

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