3(3z^2-4+1)-11(2z-1)+10=0

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Solution for 3(3z^2-4+1)-11(2z-1)+10=0 equation:


Simplifying
3(3z2 + -4 + 1) + -11(2z + -1) + 10 = 0

Reorder the terms:
3(-4 + 1 + 3z2) + -11(2z + -1) + 10 = 0

Combine like terms: -4 + 1 = -3
3(-3 + 3z2) + -11(2z + -1) + 10 = 0
(-3 * 3 + 3z2 * 3) + -11(2z + -1) + 10 = 0
(-9 + 9z2) + -11(2z + -1) + 10 = 0

Reorder the terms:
-9 + 9z2 + -11(-1 + 2z) + 10 = 0
-9 + 9z2 + (-1 * -11 + 2z * -11) + 10 = 0
-9 + 9z2 + (11 + -22z) + 10 = 0

Reorder the terms:
-9 + 11 + 10 + -22z + 9z2 = 0

Combine like terms: -9 + 11 = 2
2 + 10 + -22z + 9z2 = 0

Combine like terms: 2 + 10 = 12
12 + -22z + 9z2 = 0

Solving
12 + -22z + 9z2 = 0

Solving for variable 'z'.

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

Divide each side by '9'.
1.333333333 + -2.444444444z + z2 = 0

Move the constant term to the right:

Add '-1.333333333' to each side of the equation.
1.333333333 + -2.444444444z + -1.333333333 + z2 = 0 + -1.333333333

Reorder the terms:
1.333333333 + -1.333333333 + -2.444444444z + z2 = 0 + -1.333333333

Combine like terms: 1.333333333 + -1.333333333 = 0.000000000
0.000000000 + -2.444444444z + z2 = 0 + -1.333333333
-2.444444444z + z2 = 0 + -1.333333333

Combine like terms: 0 + -1.333333333 = -1.333333333
-2.444444444z + z2 = -1.333333333

The z term is -2.444444444z.  Take half its coefficient (-1.222222222).
Square it (1.493827160) and add it to both sides.

Add '1.493827160' to each side of the equation.
-2.444444444z + 1.493827160 + z2 = -1.333333333 + 1.493827160

Reorder the terms:
1.493827160 + -2.444444444z + z2 = -1.333333333 + 1.493827160

Combine like terms: -1.333333333 + 1.493827160 = 0.160493827
1.493827160 + -2.444444444z + z2 = 0.160493827

Factor a perfect square on the left side:
(z + -1.222222222)(z + -1.222222222) = 0.160493827

Calculate the square root of the right side: 0.400616808

Break this problem into two subproblems by setting 
(z + -1.222222222) equal to 0.400616808 and -0.400616808.

Subproblem 1

z + -1.222222222 = 0.400616808 Simplifying z + -1.222222222 = 0.400616808 Reorder the terms: -1.222222222 + z = 0.400616808 Solving -1.222222222 + z = 0.400616808 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.222222222' to each side of the equation. -1.222222222 + 1.222222222 + z = 0.400616808 + 1.222222222 Combine like terms: -1.222222222 + 1.222222222 = 0.000000000 0.000000000 + z = 0.400616808 + 1.222222222 z = 0.400616808 + 1.222222222 Combine like terms: 0.400616808 + 1.222222222 = 1.62283903 z = 1.62283903 Simplifying z = 1.62283903

Subproblem 2

z + -1.222222222 = -0.400616808 Simplifying z + -1.222222222 = -0.400616808 Reorder the terms: -1.222222222 + z = -0.400616808 Solving -1.222222222 + z = -0.400616808 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.222222222' to each side of the equation. -1.222222222 + 1.222222222 + z = -0.400616808 + 1.222222222 Combine like terms: -1.222222222 + 1.222222222 = 0.000000000 0.000000000 + z = -0.400616808 + 1.222222222 z = -0.400616808 + 1.222222222 Combine like terms: -0.400616808 + 1.222222222 = 0.821605414 z = 0.821605414 Simplifying z = 0.821605414

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.62283903, 0.821605414}

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