3z^2-18z+26=0

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Solution for 3z^2-18z+26=0 equation:


Simplifying
3z2 + -18z + 26 = 0

Reorder the terms:
26 + -18z + 3z2 = 0

Solving
26 + -18z + 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'.
8.666666667 + -6z + z2 = 0

Move the constant term to the right:

Add '-8.666666667' to each side of the equation.
8.666666667 + -6z + -8.666666667 + z2 = 0 + -8.666666667

Reorder the terms:
8.666666667 + -8.666666667 + -6z + z2 = 0 + -8.666666667

Combine like terms: 8.666666667 + -8.666666667 = 0.000000000
0.000000000 + -6z + z2 = 0 + -8.666666667
-6z + z2 = 0 + -8.666666667

Combine like terms: 0 + -8.666666667 = -8.666666667
-6z + z2 = -8.666666667

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 = -8.666666667 + 9

Reorder the terms:
9 + -6z + z2 = -8.666666667 + 9

Combine like terms: -8.666666667 + 9 = 0.333333333
9 + -6z + z2 = 0.333333333

Factor a perfect square on the left side:
(z + -3)(z + -3) = 0.333333333

Calculate the square root of the right side: 0.577350269

Break this problem into two subproblems by setting 
(z + -3) equal to 0.577350269 and -0.577350269.

Subproblem 1

z + -3 = 0.577350269 Simplifying z + -3 = 0.577350269 Reorder the terms: -3 + z = 0.577350269 Solving -3 + z = 0.577350269 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 = 0.577350269 + 3 Combine like terms: -3 + 3 = 0 0 + z = 0.577350269 + 3 z = 0.577350269 + 3 Combine like terms: 0.577350269 + 3 = 3.577350269 z = 3.577350269 Simplifying z = 3.577350269

Subproblem 2

z + -3 = -0.577350269 Simplifying z + -3 = -0.577350269 Reorder the terms: -3 + z = -0.577350269 Solving -3 + z = -0.577350269 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 = -0.577350269 + 3 Combine like terms: -3 + 3 = 0 0 + z = -0.577350269 + 3 z = -0.577350269 + 3 Combine like terms: -0.577350269 + 3 = 2.422649731 z = 2.422649731 Simplifying z = 2.422649731

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.577350269, 2.422649731}

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