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Simplifying -1z2 + z = -14 Reorder the terms: z + -1z2 = -14 Solving z + -1z2 = -14 Solving for variable 'z'. Reorder the terms: 14 + z + -1z2 = -14 + 14 Combine like terms: -14 + 14 = 0 14 + z + -1z2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -14 + -1z + z2 = 0 Move the constant term to the right: Add '14' to each side of the equation. -14 + -1z + 14 + z2 = 0 + 14 Reorder the terms: -14 + 14 + -1z + z2 = 0 + 14 Combine like terms: -14 + 14 = 0 0 + -1z + z2 = 0 + 14 -1z + z2 = 0 + 14 Combine like terms: 0 + 14 = 14 -1z + z2 = 14 The z term is z. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1z + 0.25 + z2 = 14 + 0.25 Reorder the terms: 0.25 + -1z + z2 = 14 + 0.25 Combine like terms: 14 + 0.25 = 14.25 0.25 + -1z + z2 = 14.25 Factor a perfect square on the left side: (z + 0.5)(z + 0.5) = 14.25 Calculate the square root of the right side: 3.774917218 Break this problem into two subproblems by setting (z + 0.5) equal to 3.774917218 and -3.774917218.Subproblem 1
z + 0.5 = 3.774917218 Simplifying z + 0.5 = 3.774917218 Reorder the terms: 0.5 + z = 3.774917218 Solving 0.5 + z = 3.774917218 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = 3.774917218 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 3.774917218 + -0.5 z = 3.774917218 + -0.5 Combine like terms: 3.774917218 + -0.5 = 3.274917218 z = 3.274917218 Simplifying z = 3.274917218Subproblem 2
z + 0.5 = -3.774917218 Simplifying z + 0.5 = -3.774917218 Reorder the terms: 0.5 + z = -3.774917218 Solving 0.5 + z = -3.774917218 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = -3.774917218 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -3.774917218 + -0.5 z = -3.774917218 + -0.5 Combine like terms: -3.774917218 + -0.5 = -4.274917218 z = -4.274917218 Simplifying z = -4.274917218Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.274917218, -4.274917218}
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