a^2-b^2+2bc-c^2=(a+b-c)(a-b+c)

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Solution for a^2-b^2+2bc-c^2=(a+b-c)(a-b+c) equation: 


Simplifying
a2 + -1b2 + 2bc + -1c2 = (a + b + -1c)(a + -1b + c)

Reorder the terms:
a2 + 2bc + -1b2 + -1c2 = (a + b + -1c)(a + -1b + c)

Multiply (a + b + -1c) * (a + -1b + c)
a2 + 2bc + -1b2 + -1c2 = (a(a + -1b + c) + b(a + -1b + c) + -1c * (a + -1b + c))
a2 + 2bc + -1b2 + -1c2 = ((a * a + -1b * a + c * a) + b(a + -1b + c) + -1c * (a + -1b + c))

Reorder the terms:
a2 + 2bc + -1b2 + -1c2 = ((-1ab + ac + a2) + b(a + -1b + c) + -1c * (a + -1b + c))
a2 + 2bc + -1b2 + -1c2 = ((-1ab + ac + a2) + b(a + -1b + c) + -1c * (a + -1b + c))
a2 + 2bc + -1b2 + -1c2 = (-1ab + ac + a2 + (a * b + -1b * b + c * b) + -1c * (a + -1b + c))

Reorder the terms:
a2 + 2bc + -1b2 + -1c2 = (-1ab + ac + a2 + (ab + bc + -1b2) + -1c * (a + -1b + c))
a2 + 2bc + -1b2 + -1c2 = (-1ab + ac + a2 + (ab + bc + -1b2) + -1c * (a + -1b + c))
a2 + 2bc + -1b2 + -1c2 = (-1ab + ac + a2 + ab + bc + -1b2 + (a * -1c + -1b * -1c + c * -1c))
a2 + 2bc + -1b2 + -1c2 = (-1ab + ac + a2 + ab + bc + -1b2 + (-1ac + 1bc + -1c2))

Reorder the terms:
a2 + 2bc + -1b2 + -1c2 = (-1ab + ab + ac + -1ac + a2 + bc + 1bc + -1b2 + -1c2)

Combine like terms: -1ab + ab = 0
a2 + 2bc + -1b2 + -1c2 = (0 + ac + -1ac + a2 + bc + 1bc + -1b2 + -1c2)
a2 + 2bc + -1b2 + -1c2 = (ac + -1ac + a2 + bc + 1bc + -1b2 + -1c2)

Combine like terms: ac + -1ac = 0
a2 + 2bc + -1b2 + -1c2 = (0 + a2 + bc + 1bc + -1b2 + -1c2)
a2 + 2bc + -1b2 + -1c2 = (a2 + bc + 1bc + -1b2 + -1c2)

Combine like terms: bc + 1bc = 2bc
a2 + 2bc + -1b2 + -1c2 = (a2 + 2bc + -1b2 + -1c2)

Add '-1a2' to each side of the equation.
a2 + 2bc + -1b2 + -1a2 + -1c2 = a2 + 2bc + -1b2 + -1a2 + -1c2

Reorder the terms:
a2 + -1a2 + 2bc + -1b2 + -1c2 = a2 + 2bc + -1b2 + -1a2 + -1c2

Combine like terms: a2 + -1a2 = 0
0 + 2bc + -1b2 + -1c2 = a2 + 2bc + -1b2 + -1a2 + -1c2
2bc + -1b2 + -1c2 = a2 + 2bc + -1b2 + -1a2 + -1c2

Reorder the terms:
2bc + -1b2 + -1c2 = a2 + -1a2 + 2bc + -1b2 + -1c2

Combine like terms: a2 + -1a2 = 0
2bc + -1b2 + -1c2 = 0 + 2bc + -1b2 + -1c2
2bc + -1b2 + -1c2 = 2bc + -1b2 + -1c2

Add '-2bc' to each side of the equation.
2bc + -1b2 + -2bc + -1c2 = 2bc + -1b2 + -2bc + -1c2

Reorder the terms:
2bc + -2bc + -1b2 + -1c2 = 2bc + -1b2 + -2bc + -1c2

Combine like terms: 2bc + -2bc = 0
0 + -1b2 + -1c2 = 2bc + -1b2 + -2bc + -1c2
-1b2 + -1c2 = 2bc + -1b2 + -2bc + -1c2

Reorder the terms:
-1b2 + -1c2 = 2bc + -2bc + -1b2 + -1c2

Combine like terms: 2bc + -2bc = 0
-1b2 + -1c2 = 0 + -1b2 + -1c2
-1b2 + -1c2 = -1b2 + -1c2

Add 'b2' to each side of the equation.
-1b2 + b2 + -1c2 = -1b2 + b2 + -1c2

Combine like terms: -1b2 + b2 = 0
0 + -1c2 = -1b2 + b2 + -1c2
-1c2 = -1b2 + b2 + -1c2

Combine like terms: -1b2 + b2 = 0
-1c2 = 0 + -1c2
-1c2 = -1c2

Add 'c2' to each side of the equation.
-1c2 + c2 = -1c2 + c2

Combine like terms: -1c2 + c2 = 0
0 = -1c2 + c2

Combine like terms: -1c2 + c2 = 0
0 = 0

Solving
0 = 0

Couldn't find a variable to solve for.

This equation is an identity, all real numbers are solutions.

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