3x^2-5x-75=0

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Solution for 3x^2-5x-75=0 equation: 


Simplifying
3x2 + -5x + -75 = 0

Reorder the terms:
-75 + -5x + 3x2 = 0

Solving
-75 + -5x + 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 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + -1.666666667x + 25 + x2 = 0 + 25

Reorder the terms:
-25 + 25 + -1.666666667x + x2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + -1.666666667x + x2 = 0 + 25
-1.666666667x + x2 = 0 + 25

Combine like terms: 0 + 25 = 25
-1.666666667x + x2 = 25

The x term is -1.666666667x.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667x + 0.6944444447 + x2 = 25 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = 25 + 0.6944444447

Combine like terms: 25 + 0.6944444447 = 25.6944444447
0.6944444447 + -1.666666667x + x2 = 25.6944444447

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 25.6944444447

Calculate the square root of the right side: 5.068968775

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 5.068968775 and -5.068968775.


Subproblem 1
x + -0.8333333335 = 5.068968775

Simplifying
x + -0.8333333335 = 5.068968775

Reorder the terms:
-0.8333333335 + x = 5.068968775

Solving
-0.8333333335 + x = 5.068968775

Solving for variable 'x'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + x = 5.068968775 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + x = 5.068968775 + 0.8333333335
x = 5.068968775 + 0.8333333335

Combine like terms: 5.068968775 + 0.8333333335 = 5.9023021085
x = 5.9023021085

Simplifying
x = 5.9023021085


Subproblem 2
x + -0.8333333335 = -5.068968775

Simplifying
x + -0.8333333335 = -5.068968775

Reorder the terms:
-0.8333333335 + x = -5.068968775

Solving
-0.8333333335 + x = -5.068968775

Solving for variable 'x'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + x = -5.068968775 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + x = -5.068968775 + 0.8333333335
x = -5.068968775 + 0.8333333335

Combine like terms: -5.068968775 + 0.8333333335 = -4.2356354415
x = -4.2356354415

Simplifying
x = -4.2356354415


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.9023021085, -4.2356354415}

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