-z^2+3z+12=0

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Solution for -z^2+3z+12=0 equation:


Simplifying
-1z2 + 3z + 12 = 0

Reorder the terms:
12 + 3z + -1z2 = 0

Solving
12 + 3z + -1z2 = 0

Solving for variable 'z'.

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-12 + -3z + z2 = 0

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + -3z + 12 + z2 = 0 + 12

Reorder the terms:
-12 + 12 + -3z + z2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + -3z + z2 = 0 + 12
-3z + z2 = 0 + 12

Combine like terms: 0 + 12 = 12
-3z + z2 = 12

The z term is -3z.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3z + 2.25 + z2 = 12 + 2.25

Reorder the terms:
2.25 + -3z + z2 = 12 + 2.25

Combine like terms: 12 + 2.25 = 14.25
2.25 + -3z + z2 = 14.25

Factor a perfect square on the left side:
(z + -1.5)(z + -1.5) = 14.25

Calculate the square root of the right side: 3.774917218

Break this problem into two subproblems by setting 
(z + -1.5) equal to 3.774917218 and -3.774917218.

Subproblem 1

z + -1.5 = 3.774917218 Simplifying z + -1.5 = 3.774917218 Reorder the terms: -1.5 + z = 3.774917218 Solving -1.5 + z = 3.774917218 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = 3.774917218 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = 3.774917218 + 1.5 z = 3.774917218 + 1.5 Combine like terms: 3.774917218 + 1.5 = 5.274917218 z = 5.274917218 Simplifying z = 5.274917218

Subproblem 2

z + -1.5 = -3.774917218 Simplifying z + -1.5 = -3.774917218 Reorder the terms: -1.5 + z = -3.774917218 Solving -1.5 + z = -3.774917218 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = -3.774917218 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = -3.774917218 + 1.5 z = -3.774917218 + 1.5 Combine like terms: -3.774917218 + 1.5 = -2.274917218 z = -2.274917218 Simplifying z = -2.274917218

Solution

The solution to the problem is based on the solutions from the subproblems. z = {5.274917218, -2.274917218}

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