z^2+9z=-10

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Solution for z^2+9z=-10 equation:


Simplifying
z2 + 9z = -10

Reorder the terms:
9z + z2 = -10

Solving
9z + z2 = -10

Solving for variable 'z'.

Reorder the terms:
10 + 9z + z2 = -10 + 10

Combine like terms: -10 + 10 = 0
10 + 9z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-10' to each side of the equation.
10 + 9z + -10 + z2 = 0 + -10

Reorder the terms:
10 + -10 + 9z + z2 = 0 + -10

Combine like terms: 10 + -10 = 0
0 + 9z + z2 = 0 + -10
9z + z2 = 0 + -10

Combine like terms: 0 + -10 = -10
9z + z2 = -10

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 = -10 + 20.25

Reorder the terms:
20.25 + 9z + z2 = -10 + 20.25

Combine like terms: -10 + 20.25 = 10.25
20.25 + 9z + z2 = 10.25

Factor a perfect square on the left side:
(z + 4.5)(z + 4.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(z + 4.5) equal to 3.201562119 and -3.201562119.

Subproblem 1

z + 4.5 = 3.201562119 Simplifying z + 4.5 = 3.201562119 Reorder the terms: 4.5 + z = 3.201562119 Solving 4.5 + z = 3.201562119 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 = 3.201562119 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = 3.201562119 + -4.5 z = 3.201562119 + -4.5 Combine like terms: 3.201562119 + -4.5 = -1.298437881 z = -1.298437881 Simplifying z = -1.298437881

Subproblem 2

z + 4.5 = -3.201562119 Simplifying z + 4.5 = -3.201562119 Reorder the terms: 4.5 + z = -3.201562119 Solving 4.5 + z = -3.201562119 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 = -3.201562119 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = -3.201562119 + -4.5 z = -3.201562119 + -4.5 Combine like terms: -3.201562119 + -4.5 = -7.701562119 z = -7.701562119 Simplifying z = -7.701562119

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-1.298437881, -7.701562119}

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