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Simplifying z2 + 30z = -47 Reorder the terms: 30z + z2 = -47 Solving 30z + z2 = -47 Solving for variable 'z'. Reorder the terms: 47 + 30z + z2 = -47 + 47 Combine like terms: -47 + 47 = 0 47 + 30z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-47' to each side of the equation. 47 + 30z + -47 + z2 = 0 + -47 Reorder the terms: 47 + -47 + 30z + z2 = 0 + -47 Combine like terms: 47 + -47 = 0 0 + 30z + z2 = 0 + -47 30z + z2 = 0 + -47 Combine like terms: 0 + -47 = -47 30z + z2 = -47 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 = -47 + 225 Reorder the terms: 225 + 30z + z2 = -47 + 225 Combine like terms: -47 + 225 = 178 225 + 30z + z2 = 178 Factor a perfect square on the left side: (z + 15)(z + 15) = 178 Calculate the square root of the right side: 13.341664064 Break this problem into two subproblems by setting (z + 15) equal to 13.341664064 and -13.341664064.Subproblem 1
z + 15 = 13.341664064 Simplifying z + 15 = 13.341664064 Reorder the terms: 15 + z = 13.341664064 Solving 15 + z = 13.341664064 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 = 13.341664064 + -15 Combine like terms: 15 + -15 = 0 0 + z = 13.341664064 + -15 z = 13.341664064 + -15 Combine like terms: 13.341664064 + -15 = -1.658335936 z = -1.658335936 Simplifying z = -1.658335936Subproblem 2
z + 15 = -13.341664064 Simplifying z + 15 = -13.341664064 Reorder the terms: 15 + z = -13.341664064 Solving 15 + z = -13.341664064 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 = -13.341664064 + -15 Combine like terms: 15 + -15 = 0 0 + z = -13.341664064 + -15 z = -13.341664064 + -15 Combine like terms: -13.341664064 + -15 = -28.341664064 z = -28.341664064 Simplifying z = -28.341664064Solution
The solution to the problem is based on the solutions from the subproblems. z = {-1.658335936, -28.341664064}
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