(z-13)(z+10)+z=360

Solution for (z-13)(z+10)+z=360 equation:

Simplifying
(z + -13)(z + 10) + z = 360

Reorder the terms:
(-13 + z)(z + 10) + z = 360

Reorder the terms:
(-13 + z)(10 + z) + z = 360

Multiply (-13 + z) * (10 + z)
(-13(10 + z) + z(10 + z)) + z = 360
((10 * -13 + z * -13) + z(10 + z)) + z = 360
((-130 + -13z) + z(10 + z)) + z = 360
(-130 + -13z + (10 * z + z * z)) + z = 360
(-130 + -13z + (10z + z2)) + z = 360

Combine like terms: -13z + 10z = -3z
(-130 + -3z + z2) + z = 360

Reorder the terms:
-130 + -3z + z + z2 = 360

Combine like terms: -3z + z = -2z
-130 + -2z + z2 = 360

Solving
-130 + -2z + z2 = 360

Solving for variable 'z'.

Reorder the terms:
-130 + -360 + -2z + z2 = 360 + -360

Combine like terms: -130 + -360 = -490
-490 + -2z + z2 = 360 + -360

Combine like terms: 360 + -360 = 0
-490 + -2z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '490' to each side of the equation.
-490 + -2z + 490 + z2 = 0 + 490

Reorder the terms:
-490 + 490 + -2z + z2 = 0 + 490

Combine like terms: -490 + 490 = 0
0 + -2z + z2 = 0 + 490
-2z + z2 = 0 + 490

Combine like terms: 0 + 490 = 490
-2z + z2 = 490

The z term is -2z.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2z + 1 + z2 = 490 + 1

Reorder the terms:
1 + -2z + z2 = 490 + 1

Combine like terms: 490 + 1 = 491
1 + -2z + z2 = 491

Factor a perfect square on the left side:
(z + -1)(z + -1) = 491

Calculate the square root of the right side: 22.158519806

Break this problem into two subproblems by setting 
(z + -1) equal to 22.158519806 and -22.158519806.

Subproblem 1
z + -1 = 22.158519806

Simplifying
z + -1 = 22.158519806

Reorder the terms:
-1 + z = 22.158519806

Solving
-1 + z = 22.158519806

Solving for variable 'z'.

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

Add '1' to each side of the equation.
-1 + 1 + z = 22.158519806 + 1

Combine like terms: -1 + 1 = 0
0 + z = 22.158519806 + 1
z = 22.158519806 + 1

Combine like terms: 22.158519806 + 1 = 23.158519806
z = 23.158519806

Simplifying
z = 23.158519806

Subproblem 2
z + -1 = -22.158519806

Simplifying
z + -1 = -22.158519806

Reorder the terms:
-1 + z = -22.158519806

Solving
-1 + z = -22.158519806

Solving for variable 'z'.

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

Add '1' to each side of the equation.
-1 + 1 + z = -22.158519806 + 1

Combine like terms: -1 + 1 = 0
0 + z = -22.158519806 + 1
z = -22.158519806 + 1

Combine like terms: -22.158519806 + 1 = -21.158519806
z = -21.158519806

Simplifying
z = -21.158519806

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {23.158519806, -21.158519806}