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Simplifying z2 = 6 + 3z Solving z2 = 6 + 3z Solving for variable 'z'. Reorder the terms: -6 + -3z + z2 = 6 + 3z + -6 + -3z Reorder the terms: -6 + -3z + z2 = 6 + -6 + 3z + -3z Combine like terms: 6 + -6 = 0 -6 + -3z + z2 = 0 + 3z + -3z -6 + -3z + z2 = 3z + -3z Combine like terms: 3z + -3z = 0 -6 + -3z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '6' to each side of the equation. -6 + -3z + 6 + z2 = 0 + 6 Reorder the terms: -6 + 6 + -3z + z2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -3z + z2 = 0 + 6 -3z + z2 = 0 + 6 Combine like terms: 0 + 6 = 6 -3z + z2 = 6 The z term is -3z. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3z + 2.25 + z2 = 6 + 2.25 Reorder the terms: 2.25 + -3z + z2 = 6 + 2.25 Combine like terms: 6 + 2.25 = 8.25 2.25 + -3z + z2 = 8.25 Factor a perfect square on the left side: (z + -1.5)(z + -1.5) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (z + -1.5) equal to 2.872281323 and -2.872281323.Subproblem 1
z + -1.5 = 2.872281323 Simplifying z + -1.5 = 2.872281323 Reorder the terms: -1.5 + z = 2.872281323 Solving -1.5 + z = 2.872281323 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = 2.872281323 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = 2.872281323 + 1.5 z = 2.872281323 + 1.5 Combine like terms: 2.872281323 + 1.5 = 4.372281323 z = 4.372281323 Simplifying z = 4.372281323Subproblem 2
z + -1.5 = -2.872281323 Simplifying z + -1.5 = -2.872281323 Reorder the terms: -1.5 + z = -2.872281323 Solving -1.5 + z = -2.872281323 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = -2.872281323 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = -2.872281323 + 1.5 z = -2.872281323 + 1.5 Combine like terms: -2.872281323 + 1.5 = -1.372281323 z = -1.372281323 Simplifying z = -1.372281323Solution
The solution to the problem is based on the solutions from the subproblems. z = {4.372281323, -1.372281323}
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