f(x)=a^2f(p)f(q)-f(p+q)

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Solution for f(x)=a^2f(p)f(q)-f(p+q) equation: 


Simplifying
f(x) = a2f(p) * f(q) + -1f(p + q)

Multiply f * x
fx = a2f(p) * f(q) + -1f(p + q)

Multiply a2f * p
fx = a2fp * f * q + -1f(p + q)

Multiply a2fp * f
fx = a2f2p * q + -1f(p + q)

Multiply a2f2p * q
fx = a2f2pq + -1f(p + q)
fx = a2f2pq + (p * -1f + q * -1f)
fx = a2f2pq + (-1fp + -1fq)

Solving
fx = a2f2pq + -1fp + -1fq

Solving for variable 'f'.

Reorder the terms:
-1a2f2pq + fp + fq + fx = a2f2pq + -1fp + -1fq + -1a2f2pq + fp + fq

Reorder the terms:
-1a2f2pq + fp + fq + fx = a2f2pq + -1a2f2pq + -1fp + fp + -1fq + fq

Combine like terms: a2f2pq + -1a2f2pq = 0
-1a2f2pq + fp + fq + fx = 0 + -1fp + fp + -1fq + fq
-1a2f2pq + fp + fq + fx = -1fp + fp + -1fq + fq

Combine like terms: -1fp + fp = 0
-1a2f2pq + fp + fq + fx = 0 + -1fq + fq
-1a2f2pq + fp + fq + fx = -1fq + fq

Combine like terms: -1fq + fq = 0
-1a2f2pq + fp + fq + fx = 0

Factor out the Greatest Common Factor (GCF), 'f'.
f(-1a2fpq + p + q + x) = 0


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

Simplifying
f = 0

Solving
f = 0

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

Simplifying
f = 0

Subproblem 2
Set the factor '(-1a2fpq + p + q + x)' equal to zero and attempt to solve:

Simplifying
-1a2fpq + p + q + x = 0

Solving
-1a2fpq + p + q + x = 0

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

Add '-1p' to each side of the equation.
-1a2fpq + p + q + -1p + x = 0 + -1p

Reorder the terms:
-1a2fpq + p + -1p + q + x = 0 + -1p

Combine like terms: p + -1p = 0
-1a2fpq + 0 + q + x = 0 + -1p
-1a2fpq + q + x = 0 + -1p
Remove the zero:
-1a2fpq + q + x = -1p

Add '-1q' to each side of the equation.
-1a2fpq + q + -1q + x = -1p + -1q

Combine like terms: q + -1q = 0
-1a2fpq + 0 + x = -1p + -1q
-1a2fpq + x = -1p + -1q

Add '-1x' to each side of the equation.
-1a2fpq + x + -1x = -1p + -1q + -1x

Combine like terms: x + -1x = 0
-1a2fpq + 0 = -1p + -1q + -1x
-1a2fpq = -1p + -1q + -1x

Divide each side by '-1a2pq'.
f = a-2q-1 + a-2p-1 + a-2p-1q-1x

Simplifying
f = a-2q-1 + a-2p-1 + a-2p-1q-1x

Reorder the terms:
f = a-2p-1 + a-2p-1q-1x + a-2q-1
Solution
f = {0, a-2p-1 + a-2p-1q-1x + a-2q-1}

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