z^2+20z=41

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Solution for z^2+20z=41 equation:


Simplifying
z2 + 20z = 41

Reorder the terms:
20z + z2 = 41

Solving
20z + z2 = 41

Solving for variable 'z'.

Reorder the terms:
-41 + 20z + z2 = 41 + -41

Combine like terms: 41 + -41 = 0
-41 + 20z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '41' to each side of the equation.
-41 + 20z + 41 + z2 = 0 + 41

Reorder the terms:
-41 + 41 + 20z + z2 = 0 + 41

Combine like terms: -41 + 41 = 0
0 + 20z + z2 = 0 + 41
20z + z2 = 0 + 41

Combine like terms: 0 + 41 = 41
20z + z2 = 41

The z term is 20z.  Take half its coefficient (10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
20z + 100 + z2 = 41 + 100

Reorder the terms:
100 + 20z + z2 = 41 + 100

Combine like terms: 41 + 100 = 141
100 + 20z + z2 = 141

Factor a perfect square on the left side:
(z + 10)(z + 10) = 141

Calculate the square root of the right side: 11.874342087

Break this problem into two subproblems by setting 
(z + 10) equal to 11.874342087 and -11.874342087.

Subproblem 1

z + 10 = 11.874342087 Simplifying z + 10 = 11.874342087 Reorder the terms: 10 + z = 11.874342087 Solving 10 + z = 11.874342087 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + z = 11.874342087 + -10 Combine like terms: 10 + -10 = 0 0 + z = 11.874342087 + -10 z = 11.874342087 + -10 Combine like terms: 11.874342087 + -10 = 1.874342087 z = 1.874342087 Simplifying z = 1.874342087

Subproblem 2

z + 10 = -11.874342087 Simplifying z + 10 = -11.874342087 Reorder the terms: 10 + z = -11.874342087 Solving 10 + z = -11.874342087 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + z = -11.874342087 + -10 Combine like terms: 10 + -10 = 0 0 + z = -11.874342087 + -10 z = -11.874342087 + -10 Combine like terms: -11.874342087 + -10 = -21.874342087 z = -21.874342087 Simplifying z = -21.874342087

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.874342087, -21.874342087}

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