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Simplifying z2 + 12z + 24 = 0 Reorder the terms: 24 + 12z + z2 = 0 Solving 24 + 12z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-24' to each side of the equation. 24 + 12z + -24 + z2 = 0 + -24 Reorder the terms: 24 + -24 + 12z + z2 = 0 + -24 Combine like terms: 24 + -24 = 0 0 + 12z + z2 = 0 + -24 12z + z2 = 0 + -24 Combine like terms: 0 + -24 = -24 12z + z2 = -24 The z term is 12z. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12z + 36 + z2 = -24 + 36 Reorder the terms: 36 + 12z + z2 = -24 + 36 Combine like terms: -24 + 36 = 12 36 + 12z + z2 = 12 Factor a perfect square on the left side: (z + 6)(z + 6) = 12 Calculate the square root of the right side: 3.464101615 Break this problem into two subproblems by setting (z + 6) equal to 3.464101615 and -3.464101615.Subproblem 1
z + 6 = 3.464101615 Simplifying z + 6 = 3.464101615 Reorder the terms: 6 + z = 3.464101615 Solving 6 + z = 3.464101615 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + z = 3.464101615 + -6 Combine like terms: 6 + -6 = 0 0 + z = 3.464101615 + -6 z = 3.464101615 + -6 Combine like terms: 3.464101615 + -6 = -2.535898385 z = -2.535898385 Simplifying z = -2.535898385Subproblem 2
z + 6 = -3.464101615 Simplifying z + 6 = -3.464101615 Reorder the terms: 6 + z = -3.464101615 Solving 6 + z = -3.464101615 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + z = -3.464101615 + -6 Combine like terms: 6 + -6 = 0 0 + z = -3.464101615 + -6 z = -3.464101615 + -6 Combine like terms: -3.464101615 + -6 = -9.464101615 z = -9.464101615 Simplifying z = -9.464101615Solution
The solution to the problem is based on the solutions from the subproblems. z = {-2.535898385, -9.464101615}
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