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Simplifying z2 + 12z + 36 = -2z Reorder the terms: 36 + 12z + z2 = -2z Solving 36 + 12z + z2 = -2z Solving for variable 'z'. Reorder the terms: 36 + 12z + 2z + z2 = -2z + 2z Combine like terms: 12z + 2z = 14z 36 + 14z + z2 = -2z + 2z Combine like terms: -2z + 2z = 0 36 + 14z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-36' to each side of the equation. 36 + 14z + -36 + z2 = 0 + -36 Reorder the terms: 36 + -36 + 14z + z2 = 0 + -36 Combine like terms: 36 + -36 = 0 0 + 14z + z2 = 0 + -36 14z + z2 = 0 + -36 Combine like terms: 0 + -36 = -36 14z + z2 = -36 The z term is 14z. Take half its coefficient (7). Square it (49) and add it to both sides. Add '49' to each side of the equation. 14z + 49 + z2 = -36 + 49 Reorder the terms: 49 + 14z + z2 = -36 + 49 Combine like terms: -36 + 49 = 13 49 + 14z + z2 = 13 Factor a perfect square on the left side: (z + 7)(z + 7) = 13 Calculate the square root of the right side: 3.605551275 Break this problem into two subproblems by setting (z + 7) equal to 3.605551275 and -3.605551275.Subproblem 1
z + 7 = 3.605551275 Simplifying z + 7 = 3.605551275 Reorder the terms: 7 + z = 3.605551275 Solving 7 + z = 3.605551275 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + z = 3.605551275 + -7 Combine like terms: 7 + -7 = 0 0 + z = 3.605551275 + -7 z = 3.605551275 + -7 Combine like terms: 3.605551275 + -7 = -3.394448725 z = -3.394448725 Simplifying z = -3.394448725Subproblem 2
z + 7 = -3.605551275 Simplifying z + 7 = -3.605551275 Reorder the terms: 7 + z = -3.605551275 Solving 7 + z = -3.605551275 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + z = -3.605551275 + -7 Combine like terms: 7 + -7 = 0 0 + z = -3.605551275 + -7 z = -3.605551275 + -7 Combine like terms: -3.605551275 + -7 = -10.605551275 z = -10.605551275 Simplifying z = -10.605551275Solution
The solution to the problem is based on the solutions from the subproblems. z = {-3.394448725, -10.605551275}
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