3z^2+6z=2

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Solution for 3z^2+6z=2 equation:


Simplifying
3z2 + 6z = 2

Reorder the terms:
6z + 3z2 = 2

Solving
6z + 3z2 = 2

Solving for variable 'z'.

Reorder the terms:
-2 + 6z + 3z2 = 2 + -2

Combine like terms: 2 + -2 = 0
-2 + 6z + 3z2 = 0

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
-0.6666666667 + 2z + z2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + 2z + 0.6666666667 + z2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + 2z + z2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + 2z + z2 = 0 + 0.6666666667
2z + z2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
2z + z2 = 0.6666666667

The z term is 2z.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2z + 1 + z2 = 0.6666666667 + 1

Reorder the terms:
1 + 2z + z2 = 0.6666666667 + 1

Combine like terms: 0.6666666667 + 1 = 1.6666666667
1 + 2z + z2 = 1.6666666667

Factor a perfect square on the left side:
(z + 1)(z + 1) = 1.6666666667

Calculate the square root of the right side: 1.290994449

Break this problem into two subproblems by setting 
(z + 1) equal to 1.290994449 and -1.290994449.

Subproblem 1

z + 1 = 1.290994449 Simplifying z + 1 = 1.290994449 Reorder the terms: 1 + z = 1.290994449 Solving 1 + z = 1.290994449 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + z = 1.290994449 + -1 Combine like terms: 1 + -1 = 0 0 + z = 1.290994449 + -1 z = 1.290994449 + -1 Combine like terms: 1.290994449 + -1 = 0.290994449 z = 0.290994449 Simplifying z = 0.290994449

Subproblem 2

z + 1 = -1.290994449 Simplifying z + 1 = -1.290994449 Reorder the terms: 1 + z = -1.290994449 Solving 1 + z = -1.290994449 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + z = -1.290994449 + -1 Combine like terms: 1 + -1 = 0 0 + z = -1.290994449 + -1 z = -1.290994449 + -1 Combine like terms: -1.290994449 + -1 = -2.290994449 z = -2.290994449 Simplifying z = -2.290994449

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.290994449, -2.290994449}

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