z^2+12z+24=0

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Solution for z^2+12z+24=0 equation:


Simplifying
z2 + 12z + 24 = 0

Reorder the terms:
24 + 12z + z2 = 0

Solving
24 + 12z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-24' to each side of the equation.
24 + 12z + -24 + z2 = 0 + -24

Reorder the terms:
24 + -24 + 12z + z2 = 0 + -24

Combine like terms: 24 + -24 = 0
0 + 12z + z2 = 0 + -24
12z + z2 = 0 + -24

Combine like terms: 0 + -24 = -24
12z + z2 = -24

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 = -24 + 36

Reorder the terms:
36 + 12z + z2 = -24 + 36

Combine like terms: -24 + 36 = 12
36 + 12z + z2 = 12

Factor a perfect square on the left side:
(z + 6)(z + 6) = 12

Calculate the square root of the right side: 3.464101615

Break this problem into two subproblems by setting 
(z + 6) equal to 3.464101615 and -3.464101615.

Subproblem 1

z + 6 = 3.464101615 Simplifying z + 6 = 3.464101615 Reorder the terms: 6 + z = 3.464101615 Solving 6 + z = 3.464101615 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 = 3.464101615 + -6 Combine like terms: 6 + -6 = 0 0 + z = 3.464101615 + -6 z = 3.464101615 + -6 Combine like terms: 3.464101615 + -6 = -2.535898385 z = -2.535898385 Simplifying z = -2.535898385

Subproblem 2

z + 6 = -3.464101615 Simplifying z + 6 = -3.464101615 Reorder the terms: 6 + z = -3.464101615 Solving 6 + z = -3.464101615 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 = -3.464101615 + -6 Combine like terms: 6 + -6 = 0 0 + z = -3.464101615 + -6 z = -3.464101615 + -6 Combine like terms: -3.464101615 + -6 = -9.464101615 z = -9.464101615 Simplifying z = -9.464101615

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-2.535898385, -9.464101615}

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