4x(x+1)=2x+10

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Solution for 4x(x+1)=2x+10 equation: 


Simplifying
4x(x + 1) = 2x + 10

Reorder the terms:
4x(1 + x) = 2x + 10
(1 * 4x + x * 4x) = 2x + 10
(4x + 4x2) = 2x + 10

Reorder the terms:
4x + 4x2 = 10 + 2x

Solving
4x + 4x2 = 10 + 2x

Solving for variable 'x'.

Reorder the terms:
-10 + 4x + -2x + 4x2 = 10 + 2x + -10 + -2x

Combine like terms: 4x + -2x = 2x
-10 + 2x + 4x2 = 10 + 2x + -10 + -2x

Reorder the terms:
-10 + 2x + 4x2 = 10 + -10 + 2x + -2x

Combine like terms: 10 + -10 = 0
-10 + 2x + 4x2 = 0 + 2x + -2x
-10 + 2x + 4x2 = 2x + -2x

Combine like terms: 2x + -2x = 0
-10 + 2x + 4x2 = 0

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

Ignore the factor 2.

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

Simplifying
-5 + x + 2x2 = 0

Solving
-5 + x + 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 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 = 2.5 + 0.25

Reorder the terms:
0.25 + 0.5x + x2 = 2.5 + 0.25

Combine like terms: 2.5 + 0.25 = 2.75
0.25 + 0.5x + x2 = 2.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 2.75

Calculate the square root of the right side: 1.658312395

Break this problem into two subproblems by setting 
(x + 0.5) equal to 1.658312395 and -1.658312395.


Subproblem 1
x + 0.5 = 1.658312395

Simplifying
x + 0.5 = 1.658312395

Reorder the terms:
0.5 + x = 1.658312395

Solving
0.5 + x = 1.658312395

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 = 1.658312395 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 1.658312395 + -0.5
x = 1.658312395 + -0.5

Combine like terms: 1.658312395 + -0.5 = 1.158312395
x = 1.158312395

Simplifying
x = 1.158312395


Subproblem 2
x + 0.5 = -1.658312395

Simplifying
x + 0.5 = -1.658312395

Reorder the terms:
0.5 + x = -1.658312395

Solving
0.5 + x = -1.658312395

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 = -1.658312395 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -1.658312395 + -0.5
x = -1.658312395 + -0.5

Combine like terms: -1.658312395 + -0.5 = -2.158312395
x = -2.158312395

Simplifying
x = -2.158312395


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.158312395, -2.158312395}
Solution
x = {1.158312395, -2.158312395}

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