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Simplifying (3z + -2)(2z) = 33 Reorder the terms: (-2 + 3z)(2z) = 33 Remove parenthesis around (2z) (-2 + 3z) * 2z = 33 Reorder the terms for easier multiplication: 2z(-2 + 3z) = 33 (-2 * 2z + 3z * 2z) = 33 (-4z + 6z2) = 33 Solving -4z + 6z2 = 33 Solving for variable 'z'. Reorder the terms: -33 + -4z + 6z2 = 33 + -33 Combine like terms: 33 + -33 = 0 -33 + -4z + 6z2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -5.5 + -0.6666666667z + z2 = 0 Move the constant term to the right: Add '5.5' to each side of the equation. -5.5 + -0.6666666667z + 5.5 + z2 = 0 + 5.5 Reorder the terms: -5.5 + 5.5 + -0.6666666667z + z2 = 0 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + -0.6666666667z + z2 = 0 + 5.5 -0.6666666667z + z2 = 0 + 5.5 Combine like terms: 0 + 5.5 = 5.5 -0.6666666667z + z2 = 5.5 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 = 5.5 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667z + z2 = 5.5 + 0.1111111112 Combine like terms: 5.5 + 0.1111111112 = 5.6111111112 0.1111111112 + -0.6666666667z + z2 = 5.6111111112 Factor a perfect square on the left side: (z + -0.3333333334)(z + -0.3333333334) = 5.6111111112 Calculate the square root of the right side: 2.368778401 Break this problem into two subproblems by setting (z + -0.3333333334) equal to 2.368778401 and -2.368778401.Subproblem 1
z + -0.3333333334 = 2.368778401 Simplifying z + -0.3333333334 = 2.368778401 Reorder the terms: -0.3333333334 + z = 2.368778401 Solving -0.3333333334 + z = 2.368778401 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 = 2.368778401 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + z = 2.368778401 + 0.3333333334 z = 2.368778401 + 0.3333333334 Combine like terms: 2.368778401 + 0.3333333334 = 2.7021117344 z = 2.7021117344 Simplifying z = 2.7021117344Subproblem 2
z + -0.3333333334 = -2.368778401 Simplifying z + -0.3333333334 = -2.368778401 Reorder the terms: -0.3333333334 + z = -2.368778401 Solving -0.3333333334 + z = -2.368778401 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 = -2.368778401 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + z = -2.368778401 + 0.3333333334 z = -2.368778401 + 0.3333333334 Combine like terms: -2.368778401 + 0.3333333334 = -2.0354450676 z = -2.0354450676 Simplifying z = -2.0354450676Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.7021117344, -2.0354450676}
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