3-4X(2-X)=7X(X+1)

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Solution for 3-4X(2-X)=7X(X+1) equation: 


Simplifying
3 + -4X(2 + -1X) = 7X(X + 1)
3 + (2 * -4X + -1X * -4X) = 7X(X + 1)
3 + (-8X + 4X2) = 7X(X + 1)

Reorder the terms:
3 + -8X + 4X2 = 7X(1 + X)
3 + -8X + 4X2 = (1 * 7X + X * 7X)
3 + -8X + 4X2 = (7X + 7X2)

Solving
3 + -8X + 4X2 = 7X + 7X2

Solving for variable 'X'.

Reorder the terms:
3 + -8X + -7X + 4X2 + -7X2 = 7X + 7X2 + -7X + -7X2

Combine like terms: -8X + -7X = -15X
3 + -15X + 4X2 + -7X2 = 7X + 7X2 + -7X + -7X2

Combine like terms: 4X2 + -7X2 = -3X2
3 + -15X + -3X2 = 7X + 7X2 + -7X + -7X2

Reorder the terms:
3 + -15X + -3X2 = 7X + -7X + 7X2 + -7X2

Combine like terms: 7X + -7X = 0
3 + -15X + -3X2 = 0 + 7X2 + -7X2
3 + -15X + -3X2 = 7X2 + -7X2

Combine like terms: 7X2 + -7X2 = 0
3 + -15X + -3X2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(1 + -5X + -1X2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(1 + -5X + -1X2)' equal to zero and attempt to solve:

Simplifying
1 + -5X + -1X2 = 0

Solving
1 + -5X + -1X2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-1 + 5X + X2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 5X + 1 + X2 = 0 + 1

Reorder the terms:
-1 + 1 + 5X + X2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 5X + X2 = 0 + 1
5X + X2 = 0 + 1

Combine like terms: 0 + 1 = 1
5X + X2 = 1

The X term is 5X.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5X + 6.25 + X2 = 1 + 6.25

Reorder the terms:
6.25 + 5X + X2 = 1 + 6.25

Combine like terms: 1 + 6.25 = 7.25
6.25 + 5X + X2 = 7.25

Factor a perfect square on the left side:
(X + 2.5)(X + 2.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(X + 2.5) equal to 2.692582404 and -2.692582404.


Subproblem 1
X + 2.5 = 2.692582404

Simplifying
X + 2.5 = 2.692582404

Reorder the terms:
2.5 + X = 2.692582404

Solving
2.5 + X = 2.692582404

Solving for variable 'X'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + X = 2.692582404 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + X = 2.692582404 + -2.5
X = 2.692582404 + -2.5

Combine like terms: 2.692582404 + -2.5 = 0.192582404
X = 0.192582404

Simplifying
X = 0.192582404


Subproblem 2
X + 2.5 = -2.692582404

Simplifying
X + 2.5 = -2.692582404

Reorder the terms:
2.5 + X = -2.692582404

Solving
2.5 + X = -2.692582404

Solving for variable 'X'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + X = -2.692582404 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + X = -2.692582404 + -2.5
X = -2.692582404 + -2.5

Combine like terms: -2.692582404 + -2.5 = -5.192582404
X = -5.192582404

Simplifying
X = -5.192582404


Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {0.192582404, -5.192582404}
Solution
X = {0.192582404, -5.192582404}

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