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Simplifying z(z + 9) = 1 Reorder the terms: z(9 + z) = 1 (9 * z + z * z) = 1 (9z + z2) = 1 Solving 9z + z2 = 1 Solving for variable 'z'. Reorder the terms: -1 + 9z + z2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 9z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + 9z + 1 + z2 = 0 + 1 Reorder the terms: -1 + 1 + 9z + z2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 9z + z2 = 0 + 1 9z + z2 = 0 + 1 Combine like terms: 0 + 1 = 1 9z + z2 = 1 The z term is 9z. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9z + 20.25 + z2 = 1 + 20.25 Reorder the terms: 20.25 + 9z + z2 = 1 + 20.25 Combine like terms: 1 + 20.25 = 21.25 20.25 + 9z + z2 = 21.25 Factor a perfect square on the left side: (z + 4.5)(z + 4.5) = 21.25 Calculate the square root of the right side: 4.609772229 Break this problem into two subproblems by setting (z + 4.5) equal to 4.609772229 and -4.609772229.Subproblem 1
z + 4.5 = 4.609772229 Simplifying z + 4.5 = 4.609772229 Reorder the terms: 4.5 + z = 4.609772229 Solving 4.5 + z = 4.609772229 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + z = 4.609772229 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = 4.609772229 + -4.5 z = 4.609772229 + -4.5 Combine like terms: 4.609772229 + -4.5 = 0.109772229 z = 0.109772229 Simplifying z = 0.109772229Subproblem 2
z + 4.5 = -4.609772229 Simplifying z + 4.5 = -4.609772229 Reorder the terms: 4.5 + z = -4.609772229 Solving 4.5 + z = -4.609772229 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + z = -4.609772229 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = -4.609772229 + -4.5 z = -4.609772229 + -4.5 Combine like terms: -4.609772229 + -4.5 = -9.109772229 z = -9.109772229 Simplifying z = -9.109772229Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.109772229, -9.109772229}
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