-3z^2+z+1=0

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Solution for -3z^2+z+1=0 equation:


Simplifying
-3z2 + z + 1 = 0

Reorder the terms:
1 + z + -3z2 = 0

Solving
1 + z + -3z2 = 0

Solving for variable 'z'.

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

Divide each side by '-3'.
-0.3333333333 + -0.3333333333z + z2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -0.3333333333z + 0.3333333333 + z2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -0.3333333333z + z2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -0.3333333333z + z2 = 0 + 0.3333333333
-0.3333333333z + z2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-0.3333333333z + z2 = 0.3333333333

The z term is z.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-0.3333333333z + 0.25 + z2 = 0.3333333333 + 0.25

Reorder the terms:
0.25 + -0.3333333333z + z2 = 0.3333333333 + 0.25

Combine like terms: 0.3333333333 + 0.25 = 0.5833333333
0.25 + -0.3333333333z + z2 = 0.5833333333

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 0.5833333333

Calculate the square root of the right side: 0.763762616

Break this problem into two subproblems by setting 
(z + 0.5) equal to 0.763762616 and -0.763762616.

Subproblem 1

z + 0.5 = 0.763762616 Simplifying z + 0.5 = 0.763762616 Reorder the terms: 0.5 + z = 0.763762616 Solving 0.5 + z = 0.763762616 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = 0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 0.763762616 + -0.5 z = 0.763762616 + -0.5 Combine like terms: 0.763762616 + -0.5 = 0.263762616 z = 0.263762616 Simplifying z = 0.263762616

Subproblem 2

z + 0.5 = -0.763762616 Simplifying z + 0.5 = -0.763762616 Reorder the terms: 0.5 + z = -0.763762616 Solving 0.5 + z = -0.763762616 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = -0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -0.763762616 + -0.5 z = -0.763762616 + -0.5 Combine like terms: -0.763762616 + -0.5 = -1.263762616 z = -1.263762616 Simplifying z = -1.263762616

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.263762616, -1.263762616}

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