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Simplifying z2 + 9z = 100 Reorder the terms: 9z + z2 = 100 Solving 9z + z2 = 100 Solving for variable 'z'. Reorder the terms: -100 + 9z + z2 = 100 + -100 Combine like terms: 100 + -100 = 0 -100 + 9z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '100' to each side of the equation. -100 + 9z + 100 + z2 = 0 + 100 Reorder the terms: -100 + 100 + 9z + z2 = 0 + 100 Combine like terms: -100 + 100 = 0 0 + 9z + z2 = 0 + 100 9z + z2 = 0 + 100 Combine like terms: 0 + 100 = 100 9z + z2 = 100 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 = 100 + 20.25 Reorder the terms: 20.25 + 9z + z2 = 100 + 20.25 Combine like terms: 100 + 20.25 = 120.25 20.25 + 9z + z2 = 120.25 Factor a perfect square on the left side: (z + 4.5)(z + 4.5) = 120.25 Calculate the square root of the right side: 10.9658561 Break this problem into two subproblems by setting (z + 4.5) equal to 10.9658561 and -10.9658561.Subproblem 1
z + 4.5 = 10.9658561 Simplifying z + 4.5 = 10.9658561 Reorder the terms: 4.5 + z = 10.9658561 Solving 4.5 + z = 10.9658561 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 = 10.9658561 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = 10.9658561 + -4.5 z = 10.9658561 + -4.5 Combine like terms: 10.9658561 + -4.5 = 6.4658561 z = 6.4658561 Simplifying z = 6.4658561Subproblem 2
z + 4.5 = -10.9658561 Simplifying z + 4.5 = -10.9658561 Reorder the terms: 4.5 + z = -10.9658561 Solving 4.5 + z = -10.9658561 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 = -10.9658561 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = -10.9658561 + -4.5 z = -10.9658561 + -4.5 Combine like terms: -10.9658561 + -4.5 = -15.4658561 z = -15.4658561 Simplifying z = -15.4658561Solution
The solution to the problem is based on the solutions from the subproblems. z = {6.4658561, -15.4658561}
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