12z+z^2=34

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Solution for 12z+z^2=34 equation:


Simplifying
12z + z2 = 34

Solving
12z + z2 = 34

Solving for variable 'z'.

Reorder the terms:
-34 + 12z + z2 = 34 + -34

Combine like terms: 34 + -34 = 0
-34 + 12z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '34' to each side of the equation.
-34 + 12z + 34 + z2 = 0 + 34

Reorder the terms:
-34 + 34 + 12z + z2 = 0 + 34

Combine like terms: -34 + 34 = 0
0 + 12z + z2 = 0 + 34
12z + z2 = 0 + 34

Combine like terms: 0 + 34 = 34
12z + z2 = 34

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 = 34 + 36

Reorder the terms:
36 + 12z + z2 = 34 + 36

Combine like terms: 34 + 36 = 70
36 + 12z + z2 = 70

Factor a perfect square on the left side:
(z + 6)(z + 6) = 70

Calculate the square root of the right side: 8.366600265

Break this problem into two subproblems by setting 
(z + 6) equal to 8.366600265 and -8.366600265.

Subproblem 1

z + 6 = 8.366600265 Simplifying z + 6 = 8.366600265 Reorder the terms: 6 + z = 8.366600265 Solving 6 + z = 8.366600265 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 = 8.366600265 + -6 Combine like terms: 6 + -6 = 0 0 + z = 8.366600265 + -6 z = 8.366600265 + -6 Combine like terms: 8.366600265 + -6 = 2.366600265 z = 2.366600265 Simplifying z = 2.366600265

Subproblem 2

z + 6 = -8.366600265 Simplifying z + 6 = -8.366600265 Reorder the terms: 6 + z = -8.366600265 Solving 6 + z = -8.366600265 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 = -8.366600265 + -6 Combine like terms: 6 + -6 = 0 0 + z = -8.366600265 + -6 z = -8.366600265 + -6 Combine like terms: -8.366600265 + -6 = -14.366600265 z = -14.366600265 Simplifying z = -14.366600265

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.366600265, -14.366600265}

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