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

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Solution for (3-x)(x-2)-(x-1)(x+2)=2(x+2)(x-2) equation:


Simplifying
(3 + -1x)(x + -2) + -1(x + -1)(x + 2) = 2(x + 2)(x + -2)

Reorder the terms:
(3 + -1x)(-2 + x) + -1(x + -1)(x + 2) = 2(x + 2)(x + -2)

Multiply (3 + -1x) * (-2 + x)
(3(-2 + x) + -1x * (-2 + x)) + -1(x + -1)(x + 2) = 2(x + 2)(x + -2)
((-2 * 3 + x * 3) + -1x * (-2 + x)) + -1(x + -1)(x + 2) = 2(x + 2)(x + -2)
((-6 + 3x) + -1x * (-2 + x)) + -1(x + -1)(x + 2) = 2(x + 2)(x + -2)
(-6 + 3x + (-2 * -1x + x * -1x)) + -1(x + -1)(x + 2) = 2(x + 2)(x + -2)
(-6 + 3x + (2x + -1x2)) + -1(x + -1)(x + 2) = 2(x + 2)(x + -2)

Combine like terms: 3x + 2x = 5x
(-6 + 5x + -1x2) + -1(x + -1)(x + 2) = 2(x + 2)(x + -2)

Reorder the terms:
-6 + 5x + -1x2 + -1(-1 + x)(x + 2) = 2(x + 2)(x + -2)

Reorder the terms:
-6 + 5x + -1x2 + -1(-1 + x)(2 + x) = 2(x + 2)(x + -2)

Multiply (-1 + x) * (2 + x)
-6 + 5x + -1x2 + -1(-1(2 + x) + x(2 + x)) = 2(x + 2)(x + -2)
-6 + 5x + -1x2 + -1((2 * -1 + x * -1) + x(2 + x)) = 2(x + 2)(x + -2)
-6 + 5x + -1x2 + -1((-2 + -1x) + x(2 + x)) = 2(x + 2)(x + -2)
-6 + 5x + -1x2 + -1(-2 + -1x + (2 * x + x * x)) = 2(x + 2)(x + -2)
-6 + 5x + -1x2 + -1(-2 + -1x + (2x + x2)) = 2(x + 2)(x + -2)

Combine like terms: -1x + 2x = 1x
-6 + 5x + -1x2 + -1(-2 + 1x + x2) = 2(x + 2)(x + -2)
-6 + 5x + -1x2 + (-2 * -1 + 1x * -1 + x2 * -1) = 2(x + 2)(x + -2)
-6 + 5x + -1x2 + (2 + -1x + -1x2) = 2(x + 2)(x + -2)

Reorder the terms:
-6 + 2 + 5x + -1x + -1x2 + -1x2 = 2(x + 2)(x + -2)

Combine like terms: -6 + 2 = -4
-4 + 5x + -1x + -1x2 + -1x2 = 2(x + 2)(x + -2)

Combine like terms: 5x + -1x = 4x
-4 + 4x + -1x2 + -1x2 = 2(x + 2)(x + -2)

Combine like terms: -1x2 + -1x2 = -2x2
-4 + 4x + -2x2 = 2(x + 2)(x + -2)

Reorder the terms:
-4 + 4x + -2x2 = 2(2 + x)(x + -2)

Reorder the terms:
-4 + 4x + -2x2 = 2(2 + x)(-2 + x)

Multiply (2 + x) * (-2 + x)
-4 + 4x + -2x2 = 2(2(-2 + x) + x(-2 + x))
-4 + 4x + -2x2 = 2((-2 * 2 + x * 2) + x(-2 + x))
-4 + 4x + -2x2 = 2((-4 + 2x) + x(-2 + x))
-4 + 4x + -2x2 = 2(-4 + 2x + (-2 * x + x * x))
-4 + 4x + -2x2 = 2(-4 + 2x + (-2x + x2))

Combine like terms: 2x + -2x = 0
-4 + 4x + -2x2 = 2(-4 + 0 + x2)
-4 + 4x + -2x2 = 2(-4 + x2)
-4 + 4x + -2x2 = (-4 * 2 + x2 * 2)
-4 + 4x + -2x2 = (-8 + 2x2)

Solving
-4 + 4x + -2x2 = -8 + 2x2

Solving for variable 'x'.

Reorder the terms:
-4 + 8 + 4x + -2x2 + -2x2 = -8 + 2x2 + 8 + -2x2

Combine like terms: -4 + 8 = 4
4 + 4x + -2x2 + -2x2 = -8 + 2x2 + 8 + -2x2

Combine like terms: -2x2 + -2x2 = -4x2
4 + 4x + -4x2 = -8 + 2x2 + 8 + -2x2

Reorder the terms:
4 + 4x + -4x2 = -8 + 8 + 2x2 + -2x2

Combine like terms: -8 + 8 = 0
4 + 4x + -4x2 = 0 + 2x2 + -2x2
4 + 4x + -4x2 = 2x2 + -2x2

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

Factor out the Greatest Common Factor (GCF), '4'.
4(1 + x + -1x2) = 0

Ignore the factor 4.

Subproblem 1

Set the factor '(1 + x + -1x2)' equal to zero and attempt to solve: Simplifying 1 + x + -1x2 = 0 Solving 1 + x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1 + -1x + x2 = 0 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 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. -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 = 0.618033989 x = 0.618033989 Simplifying x = 0.618033989

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 = -1.618033989 x = -1.618033989 Simplifying x = -1.618033989

Solution

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

Solution

x = {0.618033989, -1.618033989}

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