3+9z+z^2=0

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Solution for 3+9z+z^2=0 equation:


Simplifying
3 + 9z + z2 = 0

Solving
3 + 9z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 9z + -3 + z2 = 0 + -3

Reorder the terms:
3 + -3 + 9z + z2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 9z + z2 = 0 + -3
9z + z2 = 0 + -3

Combine like terms: 0 + -3 = -3
9z + z2 = -3

The z term is 9z.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9z + 20.25 + z2 = -3 + 20.25

Reorder the terms:
20.25 + 9z + z2 = -3 + 20.25

Combine like terms: -3 + 20.25 = 17.25
20.25 + 9z + z2 = 17.25

Factor a perfect square on the left side:
(z + 4.5)(z + 4.5) = 17.25

Calculate the square root of the right side: 4.153311931

Break this problem into two subproblems by setting 
(z + 4.5) equal to 4.153311931 and -4.153311931.

Subproblem 1

z + 4.5 = 4.153311931 Simplifying z + 4.5 = 4.153311931 Reorder the terms: 4.5 + z = 4.153311931 Solving 4.5 + z = 4.153311931 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + z = 4.153311931 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = 4.153311931 + -4.5 z = 4.153311931 + -4.5 Combine like terms: 4.153311931 + -4.5 = -0.346688069 z = -0.346688069 Simplifying z = -0.346688069

Subproblem 2

z + 4.5 = -4.153311931 Simplifying z + 4.5 = -4.153311931 Reorder the terms: 4.5 + z = -4.153311931 Solving 4.5 + z = -4.153311931 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + z = -4.153311931 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = -4.153311931 + -4.5 z = -4.153311931 + -4.5 Combine like terms: -4.153311931 + -4.5 = -8.653311931 z = -8.653311931 Simplifying z = -8.653311931

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.346688069, -8.653311931}

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