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Simplifying z2 + -14z + 47 = 0 Reorder the terms: 47 + -14z + z2 = 0 Solving 47 + -14z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-47' to each side of the equation. 47 + -14z + -47 + z2 = 0 + -47 Reorder the terms: 47 + -47 + -14z + z2 = 0 + -47 Combine like terms: 47 + -47 = 0 0 + -14z + z2 = 0 + -47 -14z + z2 = 0 + -47 Combine like terms: 0 + -47 = -47 -14z + z2 = -47 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 = -47 + 49 Reorder the terms: 49 + -14z + z2 = -47 + 49 Combine like terms: -47 + 49 = 2 49 + -14z + z2 = 2 Factor a perfect square on the left side: (z + -7)(z + -7) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (z + -7) equal to 1.414213562 and -1.414213562.Subproblem 1
z + -7 = 1.414213562 Simplifying z + -7 = 1.414213562 Reorder the terms: -7 + z = 1.414213562 Solving -7 + z = 1.414213562 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 = 1.414213562 + 7 Combine like terms: -7 + 7 = 0 0 + z = 1.414213562 + 7 z = 1.414213562 + 7 Combine like terms: 1.414213562 + 7 = 8.414213562 z = 8.414213562 Simplifying z = 8.414213562Subproblem 2
z + -7 = -1.414213562 Simplifying z + -7 = -1.414213562 Reorder the terms: -7 + z = -1.414213562 Solving -7 + z = -1.414213562 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 = -1.414213562 + 7 Combine like terms: -7 + 7 = 0 0 + z = -1.414213562 + 7 z = -1.414213562 + 7 Combine like terms: -1.414213562 + 7 = 5.585786438 z = 5.585786438 Simplifying z = 5.585786438Solution
The solution to the problem is based on the solutions from the subproblems. z = {8.414213562, 5.585786438}
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