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