(x-2)(x+3)=(x-2)(5-x)

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Solution for (x-2)(x+3)=(x-2)(5-x) equation:


Simplifying
(x + -2)(x + 3) = (x + -2)(5 + -1x)

Reorder the terms:
(-2 + x)(x + 3) = (x + -2)(5 + -1x)

Reorder the terms:
(-2 + x)(3 + x) = (x + -2)(5 + -1x)

Multiply (-2 + x) * (3 + x)
(-2(3 + x) + x(3 + x)) = (x + -2)(5 + -1x)
((3 * -2 + x * -2) + x(3 + x)) = (x + -2)(5 + -1x)
((-6 + -2x) + x(3 + x)) = (x + -2)(5 + -1x)
(-6 + -2x + (3 * x + x * x)) = (x + -2)(5 + -1x)
(-6 + -2x + (3x + x2)) = (x + -2)(5 + -1x)

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

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

Multiply (-2 + x) * (5 + -1x)
-6 + 1x + x2 = (-2(5 + -1x) + x(5 + -1x))
-6 + 1x + x2 = ((5 * -2 + -1x * -2) + x(5 + -1x))
-6 + 1x + x2 = ((-10 + 2x) + x(5 + -1x))
-6 + 1x + x2 = (-10 + 2x + (5 * x + -1x * x))
-6 + 1x + x2 = (-10 + 2x + (5x + -1x2))

Combine like terms: 2x + 5x = 7x
-6 + 1x + x2 = (-10 + 7x + -1x2)

Solving
-6 + 1x + x2 = -10 + 7x + -1x2

Solving for variable 'x'.

Reorder the terms:
-6 + 10 + 1x + -7x + x2 + x2 = -10 + 7x + -1x2 + 10 + -7x + x2

Combine like terms: -6 + 10 = 4
4 + 1x + -7x + x2 + x2 = -10 + 7x + -1x2 + 10 + -7x + x2

Combine like terms: 1x + -7x = -6x
4 + -6x + x2 + x2 = -10 + 7x + -1x2 + 10 + -7x + x2

Combine like terms: x2 + x2 = 2x2
4 + -6x + 2x2 = -10 + 7x + -1x2 + 10 + -7x + x2

Reorder the terms:
4 + -6x + 2x2 = -10 + 10 + 7x + -7x + -1x2 + x2

Combine like terms: -10 + 10 = 0
4 + -6x + 2x2 = 0 + 7x + -7x + -1x2 + x2
4 + -6x + 2x2 = 7x + -7x + -1x2 + x2

Combine like terms: 7x + -7x = 0
4 + -6x + 2x2 = 0 + -1x2 + x2
4 + -6x + 2x2 = -1x2 + x2

Combine like terms: -1x2 + x2 = 0
4 + -6x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(2 + -3x + x2) = 0

Factor a trinomial.
2((1 + -1x)(2 + -1x)) = 0

Ignore the factor 2.

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

Simplifying
1 + -1x = 0

Solving
1 + -1x = 0

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

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

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

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

Divide each side by '-1'.
x = 1

Simplifying
x = 1

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

Simplifying
2 + -1x = 0

Solving
2 + -1x = 0

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

Add '-2' to each side of the equation.
2 + -2 + -1x = 0 + -2

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

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

Divide each side by '-1'.
x = 2

Simplifying
x = 2
Solution
x = {1, 2}

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