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Simplifying 9z2 + -9z + -8 = 0 Reorder the terms: -8 + -9z + 9z2 = 0 Solving -8 + -9z + 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'. -0.8888888889 + -1z + z2 = 0 Move the constant term to the right: Add '0.8888888889' to each side of the equation. -0.8888888889 + -1z + 0.8888888889 + z2 = 0 + 0.8888888889 Reorder the terms: -0.8888888889 + 0.8888888889 + -1z + z2 = 0 + 0.8888888889 Combine like terms: -0.8888888889 + 0.8888888889 = 0.0000000000 0.0000000000 + -1z + z2 = 0 + 0.8888888889 -1z + z2 = 0 + 0.8888888889 Combine like terms: 0 + 0.8888888889 = 0.8888888889 -1z + z2 = 0.8888888889 The z term is -1z. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1z + 0.25 + z2 = 0.8888888889 + 0.25 Reorder the terms: 0.25 + -1z + z2 = 0.8888888889 + 0.25 Combine like terms: 0.8888888889 + 0.25 = 1.1388888889 0.25 + -1z + z2 = 1.1388888889 Factor a perfect square on the left side: (z + -0.5)(z + -0.5) = 1.1388888889 Calculate the square root of the right side: 1.067187373 Break this problem into two subproblems by setting (z + -0.5) equal to 1.067187373 and -1.067187373.Subproblem 1
z + -0.5 = 1.067187373 Simplifying z + -0.5 = 1.067187373 Reorder the terms: -0.5 + z = 1.067187373 Solving -0.5 + z = 1.067187373 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = 1.067187373 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = 1.067187373 + 0.5 z = 1.067187373 + 0.5 Combine like terms: 1.067187373 + 0.5 = 1.567187373 z = 1.567187373 Simplifying z = 1.567187373Subproblem 2
z + -0.5 = -1.067187373 Simplifying z + -0.5 = -1.067187373 Reorder the terms: -0.5 + z = -1.067187373 Solving -0.5 + z = -1.067187373 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = -1.067187373 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = -1.067187373 + 0.5 z = -1.067187373 + 0.5 Combine like terms: -1.067187373 + 0.5 = -0.567187373 z = -0.567187373 Simplifying z = -0.567187373Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.567187373, -0.567187373}
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