6z^2-9z+2=0

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Solution for 6z^2-9z+2=0 equation:


Simplifying
6z2 + -9z + 2 = 0

Reorder the terms:
2 + -9z + 6z2 = 0

Solving
2 + -9z + 6z2 = 0

Solving for variable 'z'.

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

Divide each side by '6'.
0.3333333333 + -1.5z + z2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -1.5z + -0.3333333333 + z2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -1.5z + z2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -1.5z + z2 = 0 + -0.3333333333
-1.5z + z2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-1.5z + z2 = -0.3333333333

The z term is -1.5z.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5z + 0.5625 + z2 = -0.3333333333 + 0.5625

Reorder the terms:
0.5625 + -1.5z + z2 = -0.3333333333 + 0.5625

Combine like terms: -0.3333333333 + 0.5625 = 0.2291666667
0.5625 + -1.5z + z2 = 0.2291666667

Factor a perfect square on the left side:
(z + -0.75)(z + -0.75) = 0.2291666667

Calculate the square root of the right side: 0.478713554

Break this problem into two subproblems by setting 
(z + -0.75) equal to 0.478713554 and -0.478713554.

Subproblem 1

z + -0.75 = 0.478713554 Simplifying z + -0.75 = 0.478713554 Reorder the terms: -0.75 + z = 0.478713554 Solving -0.75 + z = 0.478713554 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + z = 0.478713554 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = 0.478713554 + 0.75 z = 0.478713554 + 0.75 Combine like terms: 0.478713554 + 0.75 = 1.228713554 z = 1.228713554 Simplifying z = 1.228713554

Subproblem 2

z + -0.75 = -0.478713554 Simplifying z + -0.75 = -0.478713554 Reorder the terms: -0.75 + z = -0.478713554 Solving -0.75 + z = -0.478713554 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + z = -0.478713554 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = -0.478713554 + 0.75 z = -0.478713554 + 0.75 Combine like terms: -0.478713554 + 0.75 = 0.271286446 z = 0.271286446 Simplifying z = 0.271286446

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.228713554, 0.271286446}

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