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Simplifying z2 + 14z = -27 Reorder the terms: 14z + z2 = -27 Solving 14z + z2 = -27 Solving for variable 'z'. Reorder the terms: 27 + 14z + z2 = -27 + 27 Combine like terms: -27 + 27 = 0 27 + 14z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-27' to each side of the equation. 27 + 14z + -27 + z2 = 0 + -27 Reorder the terms: 27 + -27 + 14z + z2 = 0 + -27 Combine like terms: 27 + -27 = 0 0 + 14z + z2 = 0 + -27 14z + z2 = 0 + -27 Combine like terms: 0 + -27 = -27 14z + z2 = -27 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 = -27 + 49 Reorder the terms: 49 + 14z + z2 = -27 + 49 Combine like terms: -27 + 49 = 22 49 + 14z + z2 = 22 Factor a perfect square on the left side: (z + 7)(z + 7) = 22 Calculate the square root of the right side: 4.69041576 Break this problem into two subproblems by setting (z + 7) equal to 4.69041576 and -4.69041576.Subproblem 1
z + 7 = 4.69041576 Simplifying z + 7 = 4.69041576 Reorder the terms: 7 + z = 4.69041576 Solving 7 + z = 4.69041576 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 = 4.69041576 + -7 Combine like terms: 7 + -7 = 0 0 + z = 4.69041576 + -7 z = 4.69041576 + -7 Combine like terms: 4.69041576 + -7 = -2.30958424 z = -2.30958424 Simplifying z = -2.30958424Subproblem 2
z + 7 = -4.69041576 Simplifying z + 7 = -4.69041576 Reorder the terms: 7 + z = -4.69041576 Solving 7 + z = -4.69041576 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 = -4.69041576 + -7 Combine like terms: 7 + -7 = 0 0 + z = -4.69041576 + -7 z = -4.69041576 + -7 Combine like terms: -4.69041576 + -7 = -11.69041576 z = -11.69041576 Simplifying z = -11.69041576Solution
The solution to the problem is based on the solutions from the subproblems. z = {-2.30958424, -11.69041576}
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