z^2-12z-270=0

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Solution for z^2-12z-270=0 equation:


Simplifying
z2 + -12z + -270 = 0

Reorder the terms:
-270 + -12z + z2 = 0

Solving
-270 + -12z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '270' to each side of the equation.
-270 + -12z + 270 + z2 = 0 + 270

Reorder the terms:
-270 + 270 + -12z + z2 = 0 + 270

Combine like terms: -270 + 270 = 0
0 + -12z + z2 = 0 + 270
-12z + z2 = 0 + 270

Combine like terms: 0 + 270 = 270
-12z + z2 = 270

The z term is -12z.  Take half its coefficient (-6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
-12z + 36 + z2 = 270 + 36

Reorder the terms:
36 + -12z + z2 = 270 + 36

Combine like terms: 270 + 36 = 306
36 + -12z + z2 = 306

Factor a perfect square on the left side:
(z + -6)(z + -6) = 306

Calculate the square root of the right side: 17.492855685

Break this problem into two subproblems by setting 
(z + -6) equal to 17.492855685 and -17.492855685.

Subproblem 1

z + -6 = 17.492855685 Simplifying z + -6 = 17.492855685 Reorder the terms: -6 + z = 17.492855685 Solving -6 + z = 17.492855685 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + z = 17.492855685 + 6 Combine like terms: -6 + 6 = 0 0 + z = 17.492855685 + 6 z = 17.492855685 + 6 Combine like terms: 17.492855685 + 6 = 23.492855685 z = 23.492855685 Simplifying z = 23.492855685

Subproblem 2

z + -6 = -17.492855685 Simplifying z + -6 = -17.492855685 Reorder the terms: -6 + z = -17.492855685 Solving -6 + z = -17.492855685 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + z = -17.492855685 + 6 Combine like terms: -6 + 6 = 0 0 + z = -17.492855685 + 6 z = -17.492855685 + 6 Combine like terms: -17.492855685 + 6 = -11.492855685 z = -11.492855685 Simplifying z = -11.492855685

Solution

The solution to the problem is based on the solutions from the subproblems. z = {23.492855685, -11.492855685}

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