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Simplifying z2 + 30z + 160 = 0 Reorder the terms: 160 + 30z + z2 = 0 Solving 160 + 30z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-160' to each side of the equation. 160 + 30z + -160 + z2 = 0 + -160 Reorder the terms: 160 + -160 + 30z + z2 = 0 + -160 Combine like terms: 160 + -160 = 0 0 + 30z + z2 = 0 + -160 30z + z2 = 0 + -160 Combine like terms: 0 + -160 = -160 30z + z2 = -160 The z term is 30z. Take half its coefficient (15). Square it (225) and add it to both sides. Add '225' to each side of the equation. 30z + 225 + z2 = -160 + 225 Reorder the terms: 225 + 30z + z2 = -160 + 225 Combine like terms: -160 + 225 = 65 225 + 30z + z2 = 65 Factor a perfect square on the left side: (z + 15)(z + 15) = 65 Calculate the square root of the right side: 8.062257748 Break this problem into two subproblems by setting (z + 15) equal to 8.062257748 and -8.062257748.Subproblem 1
z + 15 = 8.062257748 Simplifying z + 15 = 8.062257748 Reorder the terms: 15 + z = 8.062257748 Solving 15 + z = 8.062257748 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + z = 8.062257748 + -15 Combine like terms: 15 + -15 = 0 0 + z = 8.062257748 + -15 z = 8.062257748 + -15 Combine like terms: 8.062257748 + -15 = -6.937742252 z = -6.937742252 Simplifying z = -6.937742252Subproblem 2
z + 15 = -8.062257748 Simplifying z + 15 = -8.062257748 Reorder the terms: 15 + z = -8.062257748 Solving 15 + z = -8.062257748 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + z = -8.062257748 + -15 Combine like terms: 15 + -15 = 0 0 + z = -8.062257748 + -15 z = -8.062257748 + -15 Combine like terms: -8.062257748 + -15 = -23.062257748 z = -23.062257748 Simplifying z = -23.062257748Solution
The solution to the problem is based on the solutions from the subproblems. z = {-6.937742252, -23.062257748}
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