(z-1)(z-1)+(z+1)(z+1)-52=0

Solution for (z-1)(z-1)+(z+1)(z+1)-52=0 equation:

Simplifying
(z + -1)(z + -1) + (z + 1)(z + 1) + -52 = 0

Reorder the terms:
(-1 + z)(z + -1) + (z + 1)(z + 1) + -52 = 0

Reorder the terms:
(-1 + z)(-1 + z) + (z + 1)(z + 1) + -52 = 0

Multiply (-1 + z) * (-1 + z)
(-1(-1 + z) + z(-1 + z)) + (z + 1)(z + 1) + -52 = 0
((-1 * -1 + z * -1) + z(-1 + z)) + (z + 1)(z + 1) + -52 = 0
((1 + -1z) + z(-1 + z)) + (z + 1)(z + 1) + -52 = 0
(1 + -1z + (-1 * z + z * z)) + (z + 1)(z + 1) + -52 = 0
(1 + -1z + (-1z + z2)) + (z + 1)(z + 1) + -52 = 0

Combine like terms: -1z + -1z = -2z
(1 + -2z + z2) + (z + 1)(z + 1) + -52 = 0

Reorder the terms:
1 + -2z + z2 + (1 + z)(z + 1) + -52 = 0

Reorder the terms:
1 + -2z + z2 + (1 + z)(1 + z) + -52 = 0

Multiply (1 + z) * (1 + z)
1 + -2z + z2 + (1(1 + z) + z(1 + z)) + -52 = 0
1 + -2z + z2 + ((1 * 1 + z * 1) + z(1 + z)) + -52 = 0
1 + -2z + z2 + ((1 + 1z) + z(1 + z)) + -52 = 0
1 + -2z + z2 + (1 + 1z + (1 * z + z * z)) + -52 = 0
1 + -2z + z2 + (1 + 1z + (1z + z2)) + -52 = 0

Combine like terms: 1z + 1z = 2z
1 + -2z + z2 + (1 + 2z + z2) + -52 = 0

Reorder the terms:
1 + 1 + -52 + -2z + 2z + z2 + z2 = 0

Combine like terms: 1 + 1 = 2
2 + -52 + -2z + 2z + z2 + z2 = 0

Combine like terms: 2 + -52 = -50
-50 + -2z + 2z + z2 + z2 = 0

Combine like terms: -2z + 2z = 0
-50 + 0 + z2 + z2 = 0
-50 + z2 + z2 = 0

Combine like terms: z2 + z2 = 2z2
-50 + 2z2 = 0

Solving
-50 + 2z2 = 0

Solving for variable 'z'.

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

Add '50' to each side of the equation.
-50 + 50 + 2z2 = 0 + 50

Combine like terms: -50 + 50 = 0
0 + 2z2 = 0 + 50
2z2 = 0 + 50

Combine like terms: 0 + 50 = 50
2z2 = 50

Divide each side by '2'.
z2 = 25

Simplifying
z2 = 25

Take the square root of each side:
z = {-5, 5}