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Simplifying 3z + -6z2 = -8 Solving 3z + -6z2 = -8 Solving for variable 'z'. Reorder the terms: 8 + 3z + -6z2 = -8 + 8 Combine like terms: -8 + 8 = 0 8 + 3z + -6z2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -1.333333333 + -0.5z + z2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + -0.5z + 1.333333333 + z2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + -0.5z + z2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + -0.5z + z2 = 0 + 1.333333333 -0.5z + z2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 -0.5z + z2 = 1.333333333 The z term is -0.5z. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5z + 0.0625 + z2 = 1.333333333 + 0.0625 Reorder the terms: 0.0625 + -0.5z + z2 = 1.333333333 + 0.0625 Combine like terms: 1.333333333 + 0.0625 = 1.395833333 0.0625 + -0.5z + z2 = 1.395833333 Factor a perfect square on the left side: (z + -0.25)(z + -0.25) = 1.395833333 Calculate the square root of the right side: 1.181453906 Break this problem into two subproblems by setting (z + -0.25) equal to 1.181453906 and -1.181453906.Subproblem 1
z + -0.25 = 1.181453906 Simplifying z + -0.25 = 1.181453906 Reorder the terms: -0.25 + z = 1.181453906 Solving -0.25 + z = 1.181453906 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + z = 1.181453906 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + z = 1.181453906 + 0.25 z = 1.181453906 + 0.25 Combine like terms: 1.181453906 + 0.25 = 1.431453906 z = 1.431453906 Simplifying z = 1.431453906Subproblem 2
z + -0.25 = -1.181453906 Simplifying z + -0.25 = -1.181453906 Reorder the terms: -0.25 + z = -1.181453906 Solving -0.25 + z = -1.181453906 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + z = -1.181453906 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + z = -1.181453906 + 0.25 z = -1.181453906 + 0.25 Combine like terms: -1.181453906 + 0.25 = -0.931453906 z = -0.931453906 Simplifying z = -0.931453906Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.431453906, -0.931453906}
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