z^2+6z=1

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Solution for z^2+6z=1 equation:


Simplifying
z2 + 6z = 1

Reorder the terms:
6z + z2 = 1

Solving
6z + z2 = 1

Solving for variable 'z'.

Reorder the terms:
-1 + 6z + z2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 6z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 6z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + 6z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 6z + z2 = 0 + 1
6z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
6z + z2 = 1

The z term is 6z.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6z + 9 + z2 = 1 + 9

Reorder the terms:
9 + 6z + z2 = 1 + 9

Combine like terms: 1 + 9 = 10
9 + 6z + z2 = 10

Factor a perfect square on the left side:
(z + 3)(z + 3) = 10

Calculate the square root of the right side: 3.16227766

Break this problem into two subproblems by setting 
(z + 3) equal to 3.16227766 and -3.16227766.

Subproblem 1

z + 3 = 3.16227766 Simplifying z + 3 = 3.16227766 Reorder the terms: 3 + z = 3.16227766 Solving 3 + z = 3.16227766 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + z = 3.16227766 + -3 Combine like terms: 3 + -3 = 0 0 + z = 3.16227766 + -3 z = 3.16227766 + -3 Combine like terms: 3.16227766 + -3 = 0.16227766 z = 0.16227766 Simplifying z = 0.16227766

Subproblem 2

z + 3 = -3.16227766 Simplifying z + 3 = -3.16227766 Reorder the terms: 3 + z = -3.16227766 Solving 3 + z = -3.16227766 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + z = -3.16227766 + -3 Combine like terms: 3 + -3 = 0 0 + z = -3.16227766 + -3 z = -3.16227766 + -3 Combine like terms: -3.16227766 + -3 = -6.16227766 z = -6.16227766 Simplifying z = -6.16227766

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.16227766, -6.16227766}

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