z^2+7z=9

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Solution for z^2+7z=9 equation:


Simplifying
z2 + 7z = 9

Reorder the terms:
7z + z2 = 9

Solving
7z + z2 = 9

Solving for variable 'z'.

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

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

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 = 21.25
12.25 + 7z + z2 = 21.25

Factor a perfect square on the left side:
(z + 3.5)(z + 3.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(z + 3.5) equal to 4.609772229 and -4.609772229.

Subproblem 1

z + 3.5 = 4.609772229 Simplifying z + 3.5 = 4.609772229 Reorder the terms: 3.5 + z = 4.609772229 Solving 3.5 + z = 4.609772229 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 = 4.609772229 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = 4.609772229 + -3.5 z = 4.609772229 + -3.5 Combine like terms: 4.609772229 + -3.5 = 1.109772229 z = 1.109772229 Simplifying z = 1.109772229

Subproblem 2

z + 3.5 = -4.609772229 Simplifying z + 3.5 = -4.609772229 Reorder the terms: 3.5 + z = -4.609772229 Solving 3.5 + z = -4.609772229 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 = -4.609772229 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = -4.609772229 + -3.5 z = -4.609772229 + -3.5 Combine like terms: -4.609772229 + -3.5 = -8.109772229 z = -8.109772229 Simplifying z = -8.109772229

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.109772229, -8.109772229}

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