10x^2-5x=25

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Solution for 10x^2-5x=25 equation: 


Simplifying
10x2 + -5x = 25

Reorder the terms:
-5x + 10x2 = 25

Solving
-5x + 10x2 = 25

Solving for variable 'x'.

Reorder the terms:
-25 + -5x + 10x2 = 25 + -25

Combine like terms: 25 + -25 = 0
-25 + -5x + 10x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-5 + -1x + 2x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-5 + -1x + 2x2)' equal to zero and attempt to solve:

Simplifying
-5 + -1x + 2x2 = 0

Solving
-5 + -1x + 2x2 = 0

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

Divide each side by '2'.
-2.5 + -0.5x + x2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + -0.5x + 2.5 + x2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + -0.5x + x2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + -0.5x + x2 = 0 + 2.5
-0.5x + x2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
-0.5x + x2 = 2.5

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 2.5 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 2.5 + 0.0625

Combine like terms: 2.5 + 0.0625 = 2.5625
0.0625 + -0.5x + x2 = 2.5625

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 2.5625

Calculate the square root of the right side: 1.600781059

Break this problem into two subproblems by setting 
(x + -0.25) equal to 1.600781059 and -1.600781059.


Subproblem 1
x + -0.25 = 1.600781059

Simplifying
x + -0.25 = 1.600781059

Reorder the terms:
-0.25 + x = 1.600781059

Solving
-0.25 + x = 1.600781059

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x = 1.600781059 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = 1.600781059 + 0.25
x = 1.600781059 + 0.25

Combine like terms: 1.600781059 + 0.25 = 1.850781059
x = 1.850781059

Simplifying
x = 1.850781059


Subproblem 2
x + -0.25 = -1.600781059

Simplifying
x + -0.25 = -1.600781059

Reorder the terms:
-0.25 + x = -1.600781059

Solving
-0.25 + x = -1.600781059

Solving for variable 'x'.

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

Add '0.25' to each side of the equation.
-0.25 + 0.25 + x = -1.600781059 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = -1.600781059 + 0.25
x = -1.600781059 + 0.25

Combine like terms: -1.600781059 + 0.25 = -1.350781059
x = -1.350781059

Simplifying
x = -1.350781059


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.850781059, -1.350781059}
Solution
x = {1.850781059, -1.350781059}

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