y^2+a^2=a^2+y^2-a(a+1)

Solution for y^2+a^2=a^2+y^2-a(a+1) equation:

Simplifying
y2 + a2 = a2 + y2 + -1a(a + 1)

Reorder the terms:
a2 + y2 = a2 + y2 + -1a(a + 1)

Reorder the terms:
a2 + y2 = a2 + y2 + -1a(1 + a)
a2 + y2 = a2 + y2 + (1 * -1a + a * -1a)
a2 + y2 = a2 + y2 + (-1a + -1a2)

Reorder the terms:
a2 + y2 = -1a + a2 + -1a2 + y2

Combine like terms: a2 + -1a2 = 0
a2 + y2 = -1a + 0 + y2
a2 + y2 = -1a + y2

Add '-1y2' to each side of the equation.
a2 + y2 + -1y2 = -1a + y2 + -1y2

Combine like terms: y2 + -1y2 = 0
a2 + 0 = -1a + y2 + -1y2
a2 = -1a + y2 + -1y2

Combine like terms: y2 + -1y2 = 0
a2 = -1a + 0
a2 = -1a

Solving
a2 = -1a

Solving for variable 'a'.

Reorder the terms:
a + a2 = -1a + a

Combine like terms: -1a + a = 0
a + a2 = 0

Factor out the Greatest Common Factor (GCF), 'a'.
a(1 + a) = 0

Subproblem 1
Set the factor 'a' equal to zero and attempt to solve:

Simplifying
a = 0

Solving
a = 0

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

Simplifying
a = 0
Subproblem 2
Set the factor '(1 + a)' equal to zero and attempt to solve:

Simplifying
1 + a = 0

Solving
1 + a = 0

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

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

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

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

Simplifying
a = -1
Solution
a = {0, -1}