(z^2)+7z+9=0

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Solution for (z^2)+7z+9=0 equation:


Simplifying
(z2) + 7z + 9 = 0
z2 + 7z + 9 = 0

Reorder the terms:
9 + 7z + z2 = 0

Solving
9 + 7z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-9' to each side of the equation.
9 + 7z + -9 + z2 = 0 + -9

Reorder the terms:
9 + -9 + 7z + z2 = 0 + -9

Combine like terms: 9 + -9 = 0
0 + 7z + z2 = 0 + -9
7z + z2 = 0 + -9

Combine like terms: 0 + -9 = -9
7z + z2 = -9

The z term is 7z.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7z + 12.25 + z2 = -9 + 12.25

Reorder the terms:
12.25 + 7z + z2 = -9 + 12.25

Combine like terms: -9 + 12.25 = 3.25
12.25 + 7z + z2 = 3.25

Factor a perfect square on the left side:
(z + 3.5)(z + 3.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(z + 3.5) equal to 1.802775638 and -1.802775638.

Subproblem 1

z + 3.5 = 1.802775638 Simplifying z + 3.5 = 1.802775638 Reorder the terms: 3.5 + z = 1.802775638 Solving 3.5 + z = 1.802775638 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = 1.802775638 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = 1.802775638 + -3.5 z = 1.802775638 + -3.5 Combine like terms: 1.802775638 + -3.5 = -1.697224362 z = -1.697224362 Simplifying z = -1.697224362

Subproblem 2

z + 3.5 = -1.802775638 Simplifying z + 3.5 = -1.802775638 Reorder the terms: 3.5 + z = -1.802775638 Solving 3.5 + z = -1.802775638 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = -1.802775638 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = -1.802775638 + -3.5 z = -1.802775638 + -3.5 Combine like terms: -1.802775638 + -3.5 = -5.302775638 z = -5.302775638 Simplifying z = -5.302775638

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-1.697224362, -5.302775638}

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