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Simplifying z2 + 9z = 32 Reorder the terms: 9z + z2 = 32 Solving 9z + z2 = 32 Solving for variable 'z'. Reorder the terms: -32 + 9z + z2 = 32 + -32 Combine like terms: 32 + -32 = 0 -32 + 9z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '32' to each side of the equation. -32 + 9z + 32 + z2 = 0 + 32 Reorder the terms: -32 + 32 + 9z + z2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + 9z + z2 = 0 + 32 9z + z2 = 0 + 32 Combine like terms: 0 + 32 = 32 9z + z2 = 32 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 = 32 + 20.25 Reorder the terms: 20.25 + 9z + z2 = 32 + 20.25 Combine like terms: 32 + 20.25 = 52.25 20.25 + 9z + z2 = 52.25 Factor a perfect square on the left side: (z + 4.5)(z + 4.5) = 52.25 Calculate the square root of the right side: 7.228416147 Break this problem into two subproblems by setting (z + 4.5) equal to 7.228416147 and -7.228416147.Subproblem 1
z + 4.5 = 7.228416147 Simplifying z + 4.5 = 7.228416147 Reorder the terms: 4.5 + z = 7.228416147 Solving 4.5 + z = 7.228416147 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 = 7.228416147 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = 7.228416147 + -4.5 z = 7.228416147 + -4.5 Combine like terms: 7.228416147 + -4.5 = 2.728416147 z = 2.728416147 Simplifying z = 2.728416147Subproblem 2
z + 4.5 = -7.228416147 Simplifying z + 4.5 = -7.228416147 Reorder the terms: 4.5 + z = -7.228416147 Solving 4.5 + z = -7.228416147 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 = -7.228416147 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = -7.228416147 + -4.5 z = -7.228416147 + -4.5 Combine like terms: -7.228416147 + -4.5 = -11.728416147 z = -11.728416147 Simplifying z = -11.728416147Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.728416147, -11.728416147}
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