z^2+9z=100

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Solution for z^2+9z=100 equation:


Simplifying
z2 + 9z = 100

Reorder the terms:
9z + z2 = 100

Solving
9z + z2 = 100

Solving for variable 'z'.

Reorder the terms:
-100 + 9z + z2 = 100 + -100

Combine like terms: 100 + -100 = 0
-100 + 9z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '100' to each side of the equation.
-100 + 9z + 100 + z2 = 0 + 100

Reorder the terms:
-100 + 100 + 9z + z2 = 0 + 100

Combine like terms: -100 + 100 = 0
0 + 9z + z2 = 0 + 100
9z + z2 = 0 + 100

Combine like terms: 0 + 100 = 100
9z + z2 = 100

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 = 100 + 20.25

Reorder the terms:
20.25 + 9z + z2 = 100 + 20.25

Combine like terms: 100 + 20.25 = 120.25
20.25 + 9z + z2 = 120.25

Factor a perfect square on the left side:
(z + 4.5)(z + 4.5) = 120.25

Calculate the square root of the right side: 10.9658561

Break this problem into two subproblems by setting 
(z + 4.5) equal to 10.9658561 and -10.9658561.

Subproblem 1

z + 4.5 = 10.9658561 Simplifying z + 4.5 = 10.9658561 Reorder the terms: 4.5 + z = 10.9658561 Solving 4.5 + z = 10.9658561 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 = 10.9658561 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = 10.9658561 + -4.5 z = 10.9658561 + -4.5 Combine like terms: 10.9658561 + -4.5 = 6.4658561 z = 6.4658561 Simplifying z = 6.4658561

Subproblem 2

z + 4.5 = -10.9658561 Simplifying z + 4.5 = -10.9658561 Reorder the terms: 4.5 + z = -10.9658561 Solving 4.5 + z = -10.9658561 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 = -10.9658561 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = -10.9658561 + -4.5 z = -10.9658561 + -4.5 Combine like terms: -10.9658561 + -4.5 = -15.4658561 z = -15.4658561 Simplifying z = -15.4658561

Solution

The solution to the problem is based on the solutions from the subproblems. z = {6.4658561, -15.4658561}

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