z^2-2.66666667z+1.777777777778=0

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Solution for z^2-2.66666667z+1.777777777778=0 equation:


Simplifying
z2 + -2.66666667z + 1.777777777778 = 0

Reorder the terms:
1.777777777778 + -2.66666667z + z2 = 0

Solving
1.777777777778 + -2.66666667z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-1.777777777778' to each side of the equation.
1.777777777778 + -2.66666667z + -1.777777777778 + z2 = 0 + -1.777777777778

Reorder the terms:
1.777777777778 + -1.777777777778 + -2.66666667z + z2 = 0 + -1.777777777778

Combine like terms: 1.777777777778 + -1.777777777778 = 0.000000000000
0.000000000000 + -2.66666667z + z2 = 0 + -1.777777777778
-2.66666667z + z2 = 0 + -1.777777777778

Combine like terms: 0 + -1.777777777778 = -1.777777777778
-2.66666667z + z2 = -1.777777777778

The z term is -2.66666667z.  Take half its coefficient (-1.333333335).
Square it (1.777777782) and add it to both sides.

Add '1.777777782' to each side of the equation.
-2.66666667z + 1.777777782 + z2 = -1.777777777778 + 1.777777782

Reorder the terms:
1.777777782 + -2.66666667z + z2 = -1.777777777778 + 1.777777782

Combine like terms: -1.777777777778 + 1.777777782 = 0.000000004222
1.777777782 + -2.66666667z + z2 = 0.000000004222

Factor a perfect square on the left side:
(z + -1.333333335)(z + -1.333333335) = 0.000000004222

Calculate the square root of the right side: 0.000064977

Break this problem into two subproblems by setting 
(z + -1.333333335) equal to 0.000064977 and -0.000064977.

Subproblem 1

z + -1.333333335 = 0.000064977 Simplifying z + -1.333333335 = 0.000064977 Reorder the terms: -1.333333335 + z = 0.000064977 Solving -1.333333335 + z = 0.000064977 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.333333335' to each side of the equation. -1.333333335 + 1.333333335 + z = 0.000064977 + 1.333333335 Combine like terms: -1.333333335 + 1.333333335 = 0.000000000 0.000000000 + z = 0.000064977 + 1.333333335 z = 0.000064977 + 1.333333335 Combine like terms: 0.000064977 + 1.333333335 = 1.333398312 z = 1.333398312 Simplifying z = 1.333398312

Subproblem 2

z + -1.333333335 = -0.000064977 Simplifying z + -1.333333335 = -0.000064977 Reorder the terms: -1.333333335 + z = -0.000064977 Solving -1.333333335 + z = -0.000064977 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.333333335' to each side of the equation. -1.333333335 + 1.333333335 + z = -0.000064977 + 1.333333335 Combine like terms: -1.333333335 + 1.333333335 = 0.000000000 0.000000000 + z = -0.000064977 + 1.333333335 z = -0.000064977 + 1.333333335 Combine like terms: -0.000064977 + 1.333333335 = 1.333268358 z = 1.333268358 Simplifying z = 1.333268358

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.333398312, 1.333268358}

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