(2x-1)(x+1)=x(x+2)

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Solution for (2x-1)(x+1)=x(x+2) equation:


Simplifying
(2x + -1)(x + 1) = x(x + 2)

Reorder the terms:
(-1 + 2x)(x + 1) = x(x + 2)

Reorder the terms:
(-1 + 2x)(1 + x) = x(x + 2)

Multiply (-1 + 2x) * (1 + x)
(-1(1 + x) + 2x * (1 + x)) = x(x + 2)
((1 * -1 + x * -1) + 2x * (1 + x)) = x(x + 2)
((-1 + -1x) + 2x * (1 + x)) = x(x + 2)
(-1 + -1x + (1 * 2x + x * 2x)) = x(x + 2)
(-1 + -1x + (2x + 2x2)) = x(x + 2)

Combine like terms: -1x + 2x = 1x
(-1 + 1x + 2x2) = x(x + 2)

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

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

Solving for variable 'x'.

Reorder the terms:
-1 + 1x + -2x + 2x2 + -1x2 = 2x + x2 + -2x + -1x2

Combine like terms: 1x + -2x = -1x
-1 + -1x + 2x2 + -1x2 = 2x + x2 + -2x + -1x2

Combine like terms: 2x2 + -1x2 = 1x2
-1 + -1x + 1x2 = 2x + x2 + -2x + -1x2

Reorder the terms:
-1 + -1x + 1x2 = 2x + -2x + x2 + -1x2

Combine like terms: 2x + -2x = 0
-1 + -1x + 1x2 = 0 + x2 + -1x2
-1 + -1x + 1x2 = x2 + -1x2

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

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -1x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -1x + x2 = 0 + 1

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

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

The x term is -1x.  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.
-1x + 0.25 + x2 = 1 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 1 + 0.25

Combine like terms: 1 + 0.25 = 1.25
0.25 + -1x + x2 = 1.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(x + -0.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

x + -0.5 = 1.118033989 Simplifying x + -0.5 = 1.118033989 Reorder the terms: -0.5 + x = 1.118033989 Solving -0.5 + x = 1.118033989 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 = 1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 1.118033989 + 0.5 x = 1.118033989 + 0.5 Combine like terms: 1.118033989 + 0.5 = 1.618033989 x = 1.618033989 Simplifying x = 1.618033989

Subproblem 2

x + -0.5 = -1.118033989 Simplifying x + -0.5 = -1.118033989 Reorder the terms: -0.5 + x = -1.118033989 Solving -0.5 + x = -1.118033989 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 = -1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -1.118033989 + 0.5 x = -1.118033989 + 0.5 Combine like terms: -1.118033989 + 0.5 = -0.618033989 x = -0.618033989 Simplifying x = -0.618033989

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.618033989, -0.618033989}

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