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Simplifying z2 + 4z = 23 Reorder the terms: 4z + z2 = 23 Solving 4z + z2 = 23 Solving for variable 'z'. Reorder the terms: -23 + 4z + z2 = 23 + -23 Combine like terms: 23 + -23 = 0 -23 + 4z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '23' to each side of the equation. -23 + 4z + 23 + z2 = 0 + 23 Reorder the terms: -23 + 23 + 4z + z2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + 4z + z2 = 0 + 23 4z + z2 = 0 + 23 Combine like terms: 0 + 23 = 23 4z + z2 = 23 The z term is 4z. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4z + 4 + z2 = 23 + 4 Reorder the terms: 4 + 4z + z2 = 23 + 4 Combine like terms: 23 + 4 = 27 4 + 4z + z2 = 27 Factor a perfect square on the left side: (z + 2)(z + 2) = 27 Calculate the square root of the right side: 5.196152423 Break this problem into two subproblems by setting (z + 2) equal to 5.196152423 and -5.196152423.Subproblem 1
z + 2 = 5.196152423 Simplifying z + 2 = 5.196152423 Reorder the terms: 2 + z = 5.196152423 Solving 2 + z = 5.196152423 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = 5.196152423 + -2 Combine like terms: 2 + -2 = 0 0 + z = 5.196152423 + -2 z = 5.196152423 + -2 Combine like terms: 5.196152423 + -2 = 3.196152423 z = 3.196152423 Simplifying z = 3.196152423Subproblem 2
z + 2 = -5.196152423 Simplifying z + 2 = -5.196152423 Reorder the terms: 2 + z = -5.196152423 Solving 2 + z = -5.196152423 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = -5.196152423 + -2 Combine like terms: 2 + -2 = 0 0 + z = -5.196152423 + -2 z = -5.196152423 + -2 Combine like terms: -5.196152423 + -2 = -7.196152423 z = -7.196152423 Simplifying z = -7.196152423Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.196152423, -7.196152423}
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