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Simplifying z2 + -13z = 32 Reorder the terms: -13z + z2 = 32 Solving -13z + z2 = 32 Solving for variable 'z'. Reorder the terms: -32 + -13z + z2 = 32 + -32 Combine like terms: 32 + -32 = 0 -32 + -13z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '32' to each side of the equation. -32 + -13z + 32 + z2 = 0 + 32 Reorder the terms: -32 + 32 + -13z + z2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + -13z + z2 = 0 + 32 -13z + z2 = 0 + 32 Combine like terms: 0 + 32 = 32 -13z + z2 = 32 The z term is -13z. Take half its coefficient (-6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. -13z + 42.25 + z2 = 32 + 42.25 Reorder the terms: 42.25 + -13z + z2 = 32 + 42.25 Combine like terms: 32 + 42.25 = 74.25 42.25 + -13z + z2 = 74.25 Factor a perfect square on the left side: (z + -6.5)(z + -6.5) = 74.25 Calculate the square root of the right side: 8.61684397 Break this problem into two subproblems by setting (z + -6.5) equal to 8.61684397 and -8.61684397.Subproblem 1
z + -6.5 = 8.61684397 Simplifying z + -6.5 = 8.61684397 Reorder the terms: -6.5 + z = 8.61684397 Solving -6.5 + z = 8.61684397 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + z = 8.61684397 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + z = 8.61684397 + 6.5 z = 8.61684397 + 6.5 Combine like terms: 8.61684397 + 6.5 = 15.11684397 z = 15.11684397 Simplifying z = 15.11684397Subproblem 2
z + -6.5 = -8.61684397 Simplifying z + -6.5 = -8.61684397 Reorder the terms: -6.5 + z = -8.61684397 Solving -6.5 + z = -8.61684397 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + z = -8.61684397 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + z = -8.61684397 + 6.5 z = -8.61684397 + 6.5 Combine like terms: -8.61684397 + 6.5 = -2.11684397 z = -2.11684397 Simplifying z = -2.11684397Solution
The solution to the problem is based on the solutions from the subproblems. z = {15.11684397, -2.11684397}
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