(x-p)(x-q)+(x-q)(x-r)+(x-r)(x-p)=0

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Solution for (x-p)(x-q)+(x-q)(x-r)+(x-r)(x-p)=0 equation: 


Simplifying
(x + -1p)(x + -1q) + (x + -1q)(x + -1r) + (x + -1r)(x + -1p) = 0

Reorder the terms:
(-1p + x)(x + -1q) + (x + -1q)(x + -1r) + (x + -1r)(x + -1p) = 0

Reorder the terms:
(-1p + x)(-1q + x) + (x + -1q)(x + -1r) + (x + -1r)(x + -1p) = 0

Multiply (-1p + x) * (-1q + x)
(-1p * (-1q + x) + x(-1q + x)) + (x + -1q)(x + -1r) + (x + -1r)(x + -1p) = 0
((-1q * -1p + x * -1p) + x(-1q + x)) + (x + -1q)(x + -1r) + (x + -1r)(x + -1p) = 0
((1pq + -1px) + x(-1q + x)) + (x + -1q)(x + -1r) + (x + -1r)(x + -1p) = 0
(1pq + -1px + (-1q * x + x * x)) + (x + -1q)(x + -1r) + (x + -1r)(x + -1p) = 0
(1pq + -1px + (-1qx + x2)) + (x + -1q)(x + -1r) + (x + -1r)(x + -1p) = 0
(1pq + -1px + -1qx + x2) + (x + -1q)(x + -1r) + (x + -1r)(x + -1p) = 0

Reorder the terms:
1pq + -1px + -1qx + x2 + (-1q + x)(x + -1r) + (x + -1r)(x + -1p) = 0

Reorder the terms:
1pq + -1px + -1qx + x2 + (-1q + x)(-1r + x) + (x + -1r)(x + -1p) = 0

Multiply (-1q + x) * (-1r + x)
1pq + -1px + -1qx + x2 + (-1q * (-1r + x) + x(-1r + x)) + (x + -1r)(x + -1p) = 0
1pq + -1px + -1qx + x2 + ((-1r * -1q + x * -1q) + x(-1r + x)) + (x + -1r)(x + -1p) = 0
1pq + -1px + -1qx + x2 + ((1qr + -1qx) + x(-1r + x)) + (x + -1r)(x + -1p) = 0
1pq + -1px + -1qx + x2 + (1qr + -1qx + (-1r * x + x * x)) + (x + -1r)(x + -1p) = 0
1pq + -1px + -1qx + x2 + (1qr + -1qx + (-1rx + x2)) + (x + -1r)(x + -1p) = 0
1pq + -1px + -1qx + x2 + (1qr + -1qx + -1rx + x2) + (x + -1r)(x + -1p) = 0

Reorder the terms:
1pq + -1px + -1qx + x2 + 1qr + -1qx + -1rx + x2 + (-1r + x)(x + -1p) = 0

Reorder the terms:
1pq + -1px + -1qx + x2 + 1qr + -1qx + -1rx + x2 + (-1r + x)(-1p + x) = 0

Multiply (-1r + x) * (-1p + x)
1pq + -1px + -1qx + x2 + 1qr + -1qx + -1rx + x2 + (-1r * (-1p + x) + x(-1p + x)) = 0
1pq + -1px + -1qx + x2 + 1qr + -1qx + -1rx + x2 + ((-1p * -1r + x * -1r) + x(-1p + x)) = 0
1pq + -1px + -1qx + x2 + 1qr + -1qx + -1rx + x2 + ((1pr + -1rx) + x(-1p + x)) = 0
1pq + -1px + -1qx + x2 + 1qr + -1qx + -1rx + x2 + (1pr + -1rx + (-1p * x + x * x)) = 0
1pq + -1px + -1qx + x2 + 1qr + -1qx + -1rx + x2 + (1pr + -1rx + (-1px + x2)) = 0

Reorder the terms:
1pq + -1px + -1qx + x2 + 1qr + -1qx + -1rx + x2 + (1pr + -1px + -1rx + x2) = 0
1pq + -1px + -1qx + x2 + 1qr + -1qx + -1rx + x2 + (1pr + -1px + -1rx + x2) = 0

Reorder the terms:
1pq + 1pr + -1px + -1px + 1qr + -1qx + -1qx + -1rx + -1rx + x2 + x2 + x2 = 0

Combine like terms: -1px + -1px = -2px
1pq + 1pr + -2px + 1qr + -1qx + -1qx + -1rx + -1rx + x2 + x2 + x2 = 0

Combine like terms: -1qx + -1qx = -2qx
1pq + 1pr + -2px + 1qr + -2qx + -1rx + -1rx + x2 + x2 + x2 = 0

Combine like terms: -1rx + -1rx = -2rx
1pq + 1pr + -2px + 1qr + -2qx + -2rx + x2 + x2 + x2 = 0

Combine like terms: x2 + x2 = 2x2
1pq + 1pr + -2px + 1qr + -2qx + -2rx + 2x2 + x2 = 0

Combine like terms: 2x2 + x2 = 3x2
1pq + 1pr + -2px + 1qr + -2qx + -2rx + 3x2 = 0

Solving
1pq + 1pr + -2px + 1qr + -2qx + -2rx + 3x2 = 0

Solving for variable 'p'.

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

Add '-1qr' to each side of the equation.
1pq + 1pr + -2px + 1qr + -2qx + -2rx + -1qr + 3x2 = 0 + -1qr

Reorder the terms:
1pq + 1pr + -2px + 1qr + -1qr + -2qx + -2rx + 3x2 = 0 + -1qr

Combine like terms: 1qr + -1qr = 0
1pq + 1pr + -2px + 0 + -2qx + -2rx + 3x2 = 0 + -1qr
1pq + 1pr + -2px + -2qx + -2rx + 3x2 = 0 + -1qr
Remove the zero:
1pq + 1pr + -2px + -2qx + -2rx + 3x2 = -1qr

Add '2qx' to each side of the equation.
1pq + 1pr + -2px + -2qx + -2rx + 2qx + 3x2 = -1qr + 2qx

Reorder the terms:
1pq + 1pr + -2px + -2qx + 2qx + -2rx + 3x2 = -1qr + 2qx

Combine like terms: -2qx + 2qx = 0
1pq + 1pr + -2px + 0 + -2rx + 3x2 = -1qr + 2qx
1pq + 1pr + -2px + -2rx + 3x2 = -1qr + 2qx

Add '2rx' to each side of the equation.
1pq + 1pr + -2px + -2rx + 2rx + 3x2 = -1qr + 2qx + 2rx

Combine like terms: -2rx + 2rx = 0
1pq + 1pr + -2px + 0 + 3x2 = -1qr + 2qx + 2rx
1pq + 1pr + -2px + 3x2 = -1qr + 2qx + 2rx

Add '-3x2' to each side of the equation.
1pq + 1pr + -2px + 3x2 + -3x2 = -1qr + 2qx + 2rx + -3x2

Combine like terms: 3x2 + -3x2 = 0
1pq + 1pr + -2px + 0 = -1qr + 2qx + 2rx + -3x2
1pq + 1pr + -2px = -1qr + 2qx + 2rx + -3x2

Reorder the terms:
1pq + 1pr + -2px + qr + -2qx + -2rx + 3x2 = -1qr + qr + 2qx + -2qx + 2rx + -2rx + -3x2 + 3x2

Combine like terms: -1qr + qr = 0
1pq + 1pr + -2px + qr + -2qx + -2rx + 3x2 = 0 + 2qx + -2qx + 2rx + -2rx + -3x2 + 3x2
1pq + 1pr + -2px + qr + -2qx + -2rx + 3x2 = 2qx + -2qx + 2rx + -2rx + -3x2 + 3x2

Combine like terms: 2qx + -2qx = 0
1pq + 1pr + -2px + qr + -2qx + -2rx + 3x2 = 0 + 2rx + -2rx + -3x2 + 3x2
1pq + 1pr + -2px + qr + -2qx + -2rx + 3x2 = 2rx + -2rx + -3x2 + 3x2

Combine like terms: 2rx + -2rx = 0
1pq + 1pr + -2px + qr + -2qx + -2rx + 3x2 = 0 + -3x2 + 3x2
1pq + 1pr + -2px + qr + -2qx + -2rx + 3x2 = -3x2 + 3x2

Combine like terms: -3x2 + 3x2 = 0
1pq + 1pr + -2px + qr + -2qx + -2rx + 3x2 = 0

The solution to this equation could not be determined.

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