7.5x^2+x-0.5=0

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Solution for 7.5x^2+x-0.5=0 equation: 


Simplifying
7.5x2 + x + -0.5 = 0

Reorder the terms:
-0.5 + x + 7.5x2 = 0

Solving
-0.5 + x + 7.5x2 = 0

Solving for variable 'x'.

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

Divide each side by '7.5'.
-0.06666666667 + 0.1333333333x + x2 = 0

Move the constant term to the right:

Add '0.06666666667' to each side of the equation.
-0.06666666667 + 0.1333333333x + 0.06666666667 + x2 = 0 + 0.06666666667

Reorder the terms:
-0.06666666667 + 0.06666666667 + 0.1333333333x + x2 = 0 + 0.06666666667

Combine like terms: -0.06666666667 + 0.06666666667 = 0.00000000000
0.00000000000 + 0.1333333333x + x2 = 0 + 0.06666666667
0.1333333333x + x2 = 0 + 0.06666666667

Combine like terms: 0 + 0.06666666667 = 0.06666666667
0.1333333333x + x2 = 0.06666666667

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.1333333333x + 0.25 + x2 = 0.06666666667 + 0.25

Reorder the terms:
0.25 + 0.1333333333x + x2 = 0.06666666667 + 0.25

Combine like terms: 0.06666666667 + 0.25 = 0.31666666667
0.25 + 0.1333333333x + x2 = 0.31666666667

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.31666666667

Calculate the square root of the right side: 0.562731434

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.562731434 and -0.562731434.


Subproblem 1
x + 0.5 = 0.562731434

Simplifying
x + 0.5 = 0.562731434

Reorder the terms:
0.5 + x = 0.562731434

Solving
0.5 + x = 0.562731434

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 0.562731434 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 0.562731434 + -0.5
x = 0.562731434 + -0.5

Combine like terms: 0.562731434 + -0.5 = 0.062731434
x = 0.062731434

Simplifying
x = 0.062731434


Subproblem 2
x + 0.5 = -0.562731434

Simplifying
x + 0.5 = -0.562731434

Reorder the terms:
0.5 + x = -0.562731434

Solving
0.5 + x = -0.562731434

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -0.562731434 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -0.562731434 + -0.5
x = -0.562731434 + -0.5

Combine like terms: -0.562731434 + -0.5 = -1.062731434
x = -1.062731434

Simplifying
x = -1.062731434


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.062731434, -1.062731434}

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