z^2+10z=1

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Solution for z^2+10z=1 equation:


Simplifying
z2 + 10z = 1

Reorder the terms:
10z + z2 = 1

Solving
10z + z2 = 1

Solving for variable 'z'.

Reorder the terms:
-1 + 10z + z2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 10z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 10z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + 10z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 10z + z2 = 0 + 1
10z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
10z + z2 = 1

The z term is 10z.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10z + 25 + z2 = 1 + 25

Reorder the terms:
25 + 10z + z2 = 1 + 25

Combine like terms: 1 + 25 = 26
25 + 10z + z2 = 26

Factor a perfect square on the left side:
(z + 5)(z + 5) = 26

Calculate the square root of the right side: 5.099019514

Break this problem into two subproblems by setting 
(z + 5) equal to 5.099019514 and -5.099019514.

Subproblem 1

z + 5 = 5.099019514 Simplifying z + 5 = 5.099019514 Reorder the terms: 5 + z = 5.099019514 Solving 5 + z = 5.099019514 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + z = 5.099019514 + -5 Combine like terms: 5 + -5 = 0 0 + z = 5.099019514 + -5 z = 5.099019514 + -5 Combine like terms: 5.099019514 + -5 = 0.099019514 z = 0.099019514 Simplifying z = 0.099019514

Subproblem 2

z + 5 = -5.099019514 Simplifying z + 5 = -5.099019514 Reorder the terms: 5 + z = -5.099019514 Solving 5 + z = -5.099019514 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + z = -5.099019514 + -5 Combine like terms: 5 + -5 = 0 0 + z = -5.099019514 + -5 z = -5.099019514 + -5 Combine like terms: -5.099019514 + -5 = -10.099019514 z = -10.099019514 Simplifying z = -10.099019514

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.099019514, -10.099019514}

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