10.5x^2+80x-75=0

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Solution for 10.5x^2+80x-75=0 equation:


Simplifying
10.5x2 + 80x + -75 = 0

Reorder the terms:
-75 + 80x + 10.5x2 = 0

Solving
-75 + 80x + 10.5x2 = 0

Solving for variable 'x'.

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

Divide each side by '10.5'.
-7.142857143 + 7.619047619x + x2 = 0

Move the constant term to the right:

Add '7.142857143' to each side of the equation.
-7.142857143 + 7.619047619x + 7.142857143 + x2 = 0 + 7.142857143

Reorder the terms:
-7.142857143 + 7.142857143 + 7.619047619x + x2 = 0 + 7.142857143

Combine like terms: -7.142857143 + 7.142857143 = 0.000000000
0.000000000 + 7.619047619x + x2 = 0 + 7.142857143
7.619047619x + x2 = 0 + 7.142857143

Combine like terms: 0 + 7.142857143 = 7.142857143
7.619047619x + x2 = 7.142857143

The x term is 7.619047619x.  Take half its coefficient (3.80952381).
Square it (14.51247166) and add it to both sides.

Add '14.51247166' to each side of the equation.
7.619047619x + 14.51247166 + x2 = 7.142857143 + 14.51247166

Reorder the terms:
14.51247166 + 7.619047619x + x2 = 7.142857143 + 14.51247166

Combine like terms: 7.142857143 + 14.51247166 = 21.655328803
14.51247166 + 7.619047619x + x2 = 21.655328803

Factor a perfect square on the left side:
(x + 3.80952381)(x + 3.80952381) = 21.655328803

Calculate the square root of the right side: 4.65352864

Break this problem into two subproblems by setting 
(x + 3.80952381) equal to 4.65352864 and -4.65352864.


Subproblem 1
x + 3.80952381 = 4.65352864

Simplifying
x + 3.80952381 = 4.65352864

Reorder the terms:
3.80952381 + x = 4.65352864

Solving
3.80952381 + x = 4.65352864

Solving for variable 'x'.

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

Add '-3.80952381' to each side of the equation.
3.80952381 + -3.80952381 + x = 4.65352864 + -3.80952381

Combine like terms: 3.80952381 + -3.80952381 = 0.00000000
0.00000000 + x = 4.65352864 + -3.80952381
x = 4.65352864 + -3.80952381

Combine like terms: 4.65352864 + -3.80952381 = 0.84400483
x = 0.84400483

Simplifying
x = 0.84400483


Subproblem 2
x + 3.80952381 = -4.65352864

Simplifying
x + 3.80952381 = -4.65352864

Reorder the terms:
3.80952381 + x = -4.65352864

Solving
3.80952381 + x = -4.65352864

Solving for variable 'x'.

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

Add '-3.80952381' to each side of the equation.
3.80952381 + -3.80952381 + x = -4.65352864 + -3.80952381

Combine like terms: 3.80952381 + -3.80952381 = 0.00000000
0.00000000 + x = -4.65352864 + -3.80952381
x = -4.65352864 + -3.80952381

Combine like terms: -4.65352864 + -3.80952381 = -8.46305245
x = -8.46305245

Simplifying
x = -8.46305245


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.84400483, -8.46305245}

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