z^2+6z=8

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Solution for z^2+6z=8 equation:


Simplifying
z2 + 6z = 8

Reorder the terms:
6z + z2 = 8

Solving
6z + z2 = 8

Solving for variable 'z'.

Reorder the terms:
-8 + 6z + z2 = 8 + -8

Combine like terms: 8 + -8 = 0
-8 + 6z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 6z + 8 + z2 = 0 + 8

Reorder the terms:
-8 + 8 + 6z + z2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + 6z + z2 = 0 + 8
6z + z2 = 0 + 8

Combine like terms: 0 + 8 = 8
6z + z2 = 8

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 = 8 + 9

Reorder the terms:
9 + 6z + z2 = 8 + 9

Combine like terms: 8 + 9 = 17
9 + 6z + z2 = 17

Factor a perfect square on the left side:
(z + 3)(z + 3) = 17

Calculate the square root of the right side: 4.123105626

Break this problem into two subproblems by setting 
(z + 3) equal to 4.123105626 and -4.123105626.

Subproblem 1

z + 3 = 4.123105626 Simplifying z + 3 = 4.123105626 Reorder the terms: 3 + z = 4.123105626 Solving 3 + z = 4.123105626 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 = 4.123105626 + -3 Combine like terms: 3 + -3 = 0 0 + z = 4.123105626 + -3 z = 4.123105626 + -3 Combine like terms: 4.123105626 + -3 = 1.123105626 z = 1.123105626 Simplifying z = 1.123105626

Subproblem 2

z + 3 = -4.123105626 Simplifying z + 3 = -4.123105626 Reorder the terms: 3 + z = -4.123105626 Solving 3 + z = -4.123105626 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 = -4.123105626 + -3 Combine like terms: 3 + -3 = 0 0 + z = -4.123105626 + -3 z = -4.123105626 + -3 Combine like terms: -4.123105626 + -3 = -7.123105626 z = -7.123105626 Simplifying z = -7.123105626

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.123105626, -7.123105626}

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