9z^2=-6z+5

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Solution for 9z^2=-6z+5 equation:


Simplifying
9z2 = -6z + 5

Reorder the terms:
9z2 = 5 + -6z

Solving
9z2 = 5 + -6z

Solving for variable 'z'.

Reorder the terms:
-5 + 6z + 9z2 = 5 + -6z + -5 + 6z

Reorder the terms:
-5 + 6z + 9z2 = 5 + -5 + -6z + 6z

Combine like terms: 5 + -5 = 0
-5 + 6z + 9z2 = 0 + -6z + 6z
-5 + 6z + 9z2 = -6z + 6z

Combine like terms: -6z + 6z = 0
-5 + 6z + 9z2 = 0

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

Divide each side by '9'.
-0.5555555556 + 0.6666666667z + z2 = 0

Move the constant term to the right:

Add '0.5555555556' to each side of the equation.
-0.5555555556 + 0.6666666667z + 0.5555555556 + z2 = 0 + 0.5555555556

Reorder the terms:
-0.5555555556 + 0.5555555556 + 0.6666666667z + z2 = 0 + 0.5555555556

Combine like terms: -0.5555555556 + 0.5555555556 = 0.0000000000
0.0000000000 + 0.6666666667z + z2 = 0 + 0.5555555556
0.6666666667z + z2 = 0 + 0.5555555556

Combine like terms: 0 + 0.5555555556 = 0.5555555556
0.6666666667z + z2 = 0.5555555556

The z term is 0.6666666667z.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667z + 0.1111111112 + z2 = 0.5555555556 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667z + z2 = 0.5555555556 + 0.1111111112

Combine like terms: 0.5555555556 + 0.1111111112 = 0.6666666668
0.1111111112 + 0.6666666667z + z2 = 0.6666666668

Factor a perfect square on the left side:
(z + 0.3333333334)(z + 0.3333333334) = 0.6666666668

Calculate the square root of the right side: 0.816496581

Break this problem into two subproblems by setting 
(z + 0.3333333334) equal to 0.816496581 and -0.816496581.

Subproblem 1

z + 0.3333333334 = 0.816496581 Simplifying z + 0.3333333334 = 0.816496581 Reorder the terms: 0.3333333334 + z = 0.816496581 Solving 0.3333333334 + z = 0.816496581 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + z = 0.816496581 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + z = 0.816496581 + -0.3333333334 z = 0.816496581 + -0.3333333334 Combine like terms: 0.816496581 + -0.3333333334 = 0.4831632476 z = 0.4831632476 Simplifying z = 0.4831632476

Subproblem 2

z + 0.3333333334 = -0.816496581 Simplifying z + 0.3333333334 = -0.816496581 Reorder the terms: 0.3333333334 + z = -0.816496581 Solving 0.3333333334 + z = -0.816496581 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + z = -0.816496581 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + z = -0.816496581 + -0.3333333334 z = -0.816496581 + -0.3333333334 Combine like terms: -0.816496581 + -0.3333333334 = -1.1498299144 z = -1.1498299144 Simplifying z = -1.1498299144

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.4831632476, -1.1498299144}

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