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Simplifying -9z2 + 10z + 5 = 0 Reorder the terms: 5 + 10z + -9z2 = 0 Solving 5 + 10z + -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'. -0.5555555556 + -1.111111111z + z2 = 0 Move the constant term to the right: Add '0.5555555556' to each side of the equation. -0.5555555556 + -1.111111111z + 0.5555555556 + z2 = 0 + 0.5555555556 Reorder the terms: -0.5555555556 + 0.5555555556 + -1.111111111z + z2 = 0 + 0.5555555556 Combine like terms: -0.5555555556 + 0.5555555556 = 0.0000000000 0.0000000000 + -1.111111111z + z2 = 0 + 0.5555555556 -1.111111111z + z2 = 0 + 0.5555555556 Combine like terms: 0 + 0.5555555556 = 0.5555555556 -1.111111111z + z2 = 0.5555555556 The z term is -1.111111111z. Take half its coefficient (-0.5555555555). Square it (0.3086419752) and add it to both sides. Add '0.3086419752' to each side of the equation. -1.111111111z + 0.3086419752 + z2 = 0.5555555556 + 0.3086419752 Reorder the terms: 0.3086419752 + -1.111111111z + z2 = 0.5555555556 + 0.3086419752 Combine like terms: 0.5555555556 + 0.3086419752 = 0.8641975308 0.3086419752 + -1.111111111z + z2 = 0.8641975308 Factor a perfect square on the left side: (z + -0.5555555555)(z + -0.5555555555) = 0.8641975308 Calculate the square root of the right side: 0.929622252 Break this problem into two subproblems by setting (z + -0.5555555555) equal to 0.929622252 and -0.929622252.Subproblem 1
z + -0.5555555555 = 0.929622252 Simplifying z + -0.5555555555 = 0.929622252 Reorder the terms: -0.5555555555 + z = 0.929622252 Solving -0.5555555555 + z = 0.929622252 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5555555555' to each side of the equation. -0.5555555555 + 0.5555555555 + z = 0.929622252 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + z = 0.929622252 + 0.5555555555 z = 0.929622252 + 0.5555555555 Combine like terms: 0.929622252 + 0.5555555555 = 1.4851778075 z = 1.4851778075 Simplifying z = 1.4851778075Subproblem 2
z + -0.5555555555 = -0.929622252 Simplifying z + -0.5555555555 = -0.929622252 Reorder the terms: -0.5555555555 + z = -0.929622252 Solving -0.5555555555 + z = -0.929622252 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5555555555' to each side of the equation. -0.5555555555 + 0.5555555555 + z = -0.929622252 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + z = -0.929622252 + 0.5555555555 z = -0.929622252 + 0.5555555555 Combine like terms: -0.929622252 + 0.5555555555 = -0.3740666965 z = -0.3740666965 Simplifying z = -0.3740666965Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.4851778075, -0.3740666965}
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