z^2+9z=32

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Solution for z^2+9z=32 equation:


Simplifying
z2 + 9z = 32

Reorder the terms:
9z + z2 = 32

Solving
9z + z2 = 32

Solving for variable 'z'.

Reorder the terms:
-32 + 9z + z2 = 32 + -32

Combine like terms: 32 + -32 = 0
-32 + 9z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '32' to each side of the equation.
-32 + 9z + 32 + z2 = 0 + 32

Reorder the terms:
-32 + 32 + 9z + z2 = 0 + 32

Combine like terms: -32 + 32 = 0
0 + 9z + z2 = 0 + 32
9z + z2 = 0 + 32

Combine like terms: 0 + 32 = 32
9z + z2 = 32

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 = 32 + 20.25

Reorder the terms:
20.25 + 9z + z2 = 32 + 20.25

Combine like terms: 32 + 20.25 = 52.25
20.25 + 9z + z2 = 52.25

Factor a perfect square on the left side:
(z + 4.5)(z + 4.5) = 52.25

Calculate the square root of the right side: 7.228416147

Break this problem into two subproblems by setting 
(z + 4.5) equal to 7.228416147 and -7.228416147.

Subproblem 1

z + 4.5 = 7.228416147 Simplifying z + 4.5 = 7.228416147 Reorder the terms: 4.5 + z = 7.228416147 Solving 4.5 + z = 7.228416147 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 = 7.228416147 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = 7.228416147 + -4.5 z = 7.228416147 + -4.5 Combine like terms: 7.228416147 + -4.5 = 2.728416147 z = 2.728416147 Simplifying z = 2.728416147

Subproblem 2

z + 4.5 = -7.228416147 Simplifying z + 4.5 = -7.228416147 Reorder the terms: 4.5 + z = -7.228416147 Solving 4.5 + z = -7.228416147 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 = -7.228416147 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = -7.228416147 + -4.5 z = -7.228416147 + -4.5 Combine like terms: -7.228416147 + -4.5 = -11.728416147 z = -11.728416147 Simplifying z = -11.728416147

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.728416147, -11.728416147}

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