x^2*10+5x=1000

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Solution for x^2*10+5x=1000 equation:


Simplifying
x2 * 10 + 5x = 1000

Reorder the terms for easier multiplication:
10x2 + 5x = 1000

Reorder the terms:
5x + 10x2 = 1000

Solving
5x + 10x2 = 1000

Solving for variable 'x'.

Reorder the terms:
-1000 + 5x + 10x2 = 1000 + -1000

Combine like terms: 1000 + -1000 = 0
-1000 + 5x + 10x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-200 + x + 2x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-200 + x + 2x2)' equal to zero and attempt to solve:

Simplifying
-200 + x + 2x2 = 0

Solving
-200 + x + 2x2 = 0

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

Divide each side by '2'.
-100 + 0.5x + x2 = 0

Move the constant term to the right:

Add '100' to each side of the equation.
-100 + 0.5x + 100 + x2 = 0 + 100

Reorder the terms:
-100 + 100 + 0.5x + x2 = 0 + 100

Combine like terms: -100 + 100 = 0
0 + 0.5x + x2 = 0 + 100
0.5x + x2 = 0 + 100

Combine like terms: 0 + 100 = 100
0.5x + x2 = 100

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.
0.5x + 0.25 + x2 = 100 + 0.25

Reorder the terms:
0.25 + 0.5x + x2 = 100 + 0.25

Combine like terms: 100 + 0.25 = 100.25
0.25 + 0.5x + x2 = 100.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 100.25

Calculate the square root of the right side: 10.012492197

Break this problem into two subproblems by setting 
(x + 0.5) equal to 10.012492197 and -10.012492197.


Subproblem 1
x + 0.5 = 10.012492197

Simplifying
x + 0.5 = 10.012492197

Reorder the terms:
0.5 + x = 10.012492197

Solving
0.5 + x = 10.012492197

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 = 10.012492197 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 10.012492197 + -0.5
x = 10.012492197 + -0.5

Combine like terms: 10.012492197 + -0.5 = 9.512492197
x = 9.512492197

Simplifying
x = 9.512492197


Subproblem 2
x + 0.5 = -10.012492197

Simplifying
x + 0.5 = -10.012492197

Reorder the terms:
0.5 + x = -10.012492197

Solving
0.5 + x = -10.012492197

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 = -10.012492197 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -10.012492197 + -0.5
x = -10.012492197 + -0.5

Combine like terms: -10.012492197 + -0.5 = -10.512492197
x = -10.512492197

Simplifying
x = -10.512492197


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {9.512492197, -10.512492197}
Solution
x = {9.512492197, -10.512492197}

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