3z^2-9z+5=0

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Solution for 3z^2-9z+5=0 equation:


Simplifying
3z2 + -9z + 5 = 0

Reorder the terms:
5 + -9z + 3z2 = 0

Solving
5 + -9z + 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'.
1.666666667 + -3z + z2 = 0

Move the constant term to the right:

Add '-1.666666667' to each side of the equation.
1.666666667 + -3z + -1.666666667 + z2 = 0 + -1.666666667

Reorder the terms:
1.666666667 + -1.666666667 + -3z + z2 = 0 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + -3z + z2 = 0 + -1.666666667
-3z + z2 = 0 + -1.666666667

Combine like terms: 0 + -1.666666667 = -1.666666667
-3z + z2 = -1.666666667

The z term is -3z.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3z + 2.25 + z2 = -1.666666667 + 2.25

Reorder the terms:
2.25 + -3z + z2 = -1.666666667 + 2.25

Combine like terms: -1.666666667 + 2.25 = 0.583333333
2.25 + -3z + z2 = 0.583333333

Factor a perfect square on the left side:
(z + -1.5)(z + -1.5) = 0.583333333

Calculate the square root of the right side: 0.763762616

Break this problem into two subproblems by setting 
(z + -1.5) equal to 0.763762616 and -0.763762616.

Subproblem 1

z + -1.5 = 0.763762616 Simplifying z + -1.5 = 0.763762616 Reorder the terms: -1.5 + z = 0.763762616 Solving -1.5 + z = 0.763762616 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = 0.763762616 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = 0.763762616 + 1.5 z = 0.763762616 + 1.5 Combine like terms: 0.763762616 + 1.5 = 2.263762616 z = 2.263762616 Simplifying z = 2.263762616

Subproblem 2

z + -1.5 = -0.763762616 Simplifying z + -1.5 = -0.763762616 Reorder the terms: -1.5 + z = -0.763762616 Solving -1.5 + z = -0.763762616 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = -0.763762616 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = -0.763762616 + 1.5 z = -0.763762616 + 1.5 Combine like terms: -0.763762616 + 1.5 = 0.736237384 z = 0.736237384 Simplifying z = 0.736237384

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.263762616, 0.736237384}

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