4x^2+90x-1000=0

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Solution for 4x^2+90x-1000=0 equation: 


Simplifying
4x2 + 90x + -1000 = 0

Reorder the terms:
-1000 + 90x + 4x2 = 0

Solving
-1000 + 90x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-500 + 45x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-500 + 45x + 2x2)' equal to zero and attempt to solve:

Simplifying
-500 + 45x + 2x2 = 0

Solving
-500 + 45x + 2x2 = 0

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

Divide each side by '2'.
-250 + 22.5x + x2 = 0

Move the constant term to the right:

Add '250' to each side of the equation.
-250 + 22.5x + 250 + x2 = 0 + 250

Reorder the terms:
-250 + 250 + 22.5x + x2 = 0 + 250

Combine like terms: -250 + 250 = 0
0 + 22.5x + x2 = 0 + 250
22.5x + x2 = 0 + 250

Combine like terms: 0 + 250 = 250
22.5x + x2 = 250

The x term is 22.5x.  Take half its coefficient (11.25).
Square it (126.5625) and add it to both sides.

Add '126.5625' to each side of the equation.
22.5x + 126.5625 + x2 = 250 + 126.5625

Reorder the terms:
126.5625 + 22.5x + x2 = 250 + 126.5625

Combine like terms: 250 + 126.5625 = 376.5625
126.5625 + 22.5x + x2 = 376.5625

Factor a perfect square on the left side:
(x + 11.25)(x + 11.25) = 376.5625

Calculate the square root of the right side: 19.40521837

Break this problem into two subproblems by setting 
(x + 11.25) equal to 19.40521837 and -19.40521837.


Subproblem 1
x + 11.25 = 19.40521837

Simplifying
x + 11.25 = 19.40521837

Reorder the terms:
11.25 + x = 19.40521837

Solving
11.25 + x = 19.40521837

Solving for variable 'x'.

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

Add '-11.25' to each side of the equation.
11.25 + -11.25 + x = 19.40521837 + -11.25

Combine like terms: 11.25 + -11.25 = 0.00
0.00 + x = 19.40521837 + -11.25
x = 19.40521837 + -11.25

Combine like terms: 19.40521837 + -11.25 = 8.15521837
x = 8.15521837

Simplifying
x = 8.15521837


Subproblem 2
x + 11.25 = -19.40521837

Simplifying
x + 11.25 = -19.40521837

Reorder the terms:
11.25 + x = -19.40521837

Solving
11.25 + x = -19.40521837

Solving for variable 'x'.

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

Add '-11.25' to each side of the equation.
11.25 + -11.25 + x = -19.40521837 + -11.25

Combine like terms: 11.25 + -11.25 = 0.00
0.00 + x = -19.40521837 + -11.25
x = -19.40521837 + -11.25

Combine like terms: -19.40521837 + -11.25 = -30.65521837
x = -30.65521837

Simplifying
x = -30.65521837


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {8.15521837, -30.65521837}
Solution
x = {8.15521837, -30.65521837}

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