x(4-x)-(5-x)(x-5)=x(x+1)

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Solution for x(4-x)-(5-x)(x-5)=x(x+1) equation:


Simplifying
x(4 + -1x) + -1(5 + -1x)(x + -5) = x(x + 1)
(4 * x + -1x * x) + -1(5 + -1x)(x + -5) = x(x + 1)
(4x + -1x2) + -1(5 + -1x)(x + -5) = x(x + 1)

Reorder the terms:
4x + -1x2 + -1(5 + -1x)(-5 + x) = x(x + 1)

Multiply (5 + -1x) * (-5 + x)
4x + -1x2 + -1(5(-5 + x) + -1x * (-5 + x)) = x(x + 1)
4x + -1x2 + -1((-5 * 5 + x * 5) + -1x * (-5 + x)) = x(x + 1)
4x + -1x2 + -1((-25 + 5x) + -1x * (-5 + x)) = x(x + 1)
4x + -1x2 + -1(-25 + 5x + (-5 * -1x + x * -1x)) = x(x + 1)
4x + -1x2 + -1(-25 + 5x + (5x + -1x2)) = x(x + 1)

Combine like terms: 5x + 5x = 10x
4x + -1x2 + -1(-25 + 10x + -1x2) = x(x + 1)
4x + -1x2 + (-25 * -1 + 10x * -1 + -1x2 * -1) = x(x + 1)
4x + -1x2 + (25 + -10x + 1x2) = x(x + 1)

Reorder the terms:
25 + 4x + -10x + -1x2 + 1x2 = x(x + 1)

Combine like terms: 4x + -10x = -6x
25 + -6x + -1x2 + 1x2 = x(x + 1)

Combine like terms: -1x2 + 1x2 = 0
25 + -6x + 0 = x(x + 1)
25 + -6x = x(x + 1)

Reorder the terms:
25 + -6x = x(1 + x)
25 + -6x = (1 * x + x * x)
25 + -6x = (1x + x2)

Solving
25 + -6x = 1x + x2

Solving for variable 'x'.

Combine like terms: -6x + -1x = -7x
25 + -7x + -1x2 = 1x + x2 + -1x + -1x2

Reorder the terms:
25 + -7x + -1x2 = 1x + -1x + x2 + -1x2

Combine like terms: 1x + -1x = 0
25 + -7x + -1x2 = 0 + x2 + -1x2
25 + -7x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
25 + -7x + -1x2 = 0

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

Divide each side by '-1'.
-25 + 7x + x2 = 0

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + 7x + 25 + x2 = 0 + 25

Reorder the terms:
-25 + 25 + 7x + x2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + 7x + x2 = 0 + 25
7x + x2 = 0 + 25

Combine like terms: 0 + 25 = 25
7x + x2 = 25

The x term is 7x.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7x + 12.25 + x2 = 25 + 12.25

Reorder the terms:
12.25 + 7x + x2 = 25 + 12.25

Combine like terms: 25 + 12.25 = 37.25
12.25 + 7x + x2 = 37.25

Factor a perfect square on the left side:
(x + 3.5)(x + 3.5) = 37.25

Calculate the square root of the right side: 6.103277808

Break this problem into two subproblems by setting 
(x + 3.5) equal to 6.103277808 and -6.103277808.

Subproblem 1

x + 3.5 = 6.103277808 Simplifying x + 3.5 = 6.103277808 Reorder the terms: 3.5 + x = 6.103277808 Solving 3.5 + x = 6.103277808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 6.103277808 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 6.103277808 + -3.5 x = 6.103277808 + -3.5 Combine like terms: 6.103277808 + -3.5 = 2.603277808 x = 2.603277808 Simplifying x = 2.603277808

Subproblem 2

x + 3.5 = -6.103277808 Simplifying x + 3.5 = -6.103277808 Reorder the terms: 3.5 + x = -6.103277808 Solving 3.5 + x = -6.103277808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -6.103277808 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -6.103277808 + -3.5 x = -6.103277808 + -3.5 Combine like terms: -6.103277808 + -3.5 = -9.603277808 x = -9.603277808 Simplifying x = -9.603277808

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.603277808, -9.603277808}

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