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Simplifying z2 + 3z + -15 = 0 Reorder the terms: -15 + 3z + z2 = 0 Solving -15 + 3z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '15' to each side of the equation. -15 + 3z + 15 + z2 = 0 + 15 Reorder the terms: -15 + 15 + 3z + z2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + 3z + z2 = 0 + 15 3z + z2 = 0 + 15 Combine like terms: 0 + 15 = 15 3z + z2 = 15 The z term is 3z. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3z + 2.25 + z2 = 15 + 2.25 Reorder the terms: 2.25 + 3z + z2 = 15 + 2.25 Combine like terms: 15 + 2.25 = 17.25 2.25 + 3z + z2 = 17.25 Factor a perfect square on the left side: (z + 1.5)(z + 1.5) = 17.25 Calculate the square root of the right side: 4.153311931 Break this problem into two subproblems by setting (z + 1.5) equal to 4.153311931 and -4.153311931.Subproblem 1
z + 1.5 = 4.153311931 Simplifying z + 1.5 = 4.153311931 Reorder the terms: 1.5 + z = 4.153311931 Solving 1.5 + z = 4.153311931 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + z = 4.153311931 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + z = 4.153311931 + -1.5 z = 4.153311931 + -1.5 Combine like terms: 4.153311931 + -1.5 = 2.653311931 z = 2.653311931 Simplifying z = 2.653311931Subproblem 2
z + 1.5 = -4.153311931 Simplifying z + 1.5 = -4.153311931 Reorder the terms: 1.5 + z = -4.153311931 Solving 1.5 + z = -4.153311931 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + z = -4.153311931 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + z = -4.153311931 + -1.5 z = -4.153311931 + -1.5 Combine like terms: -4.153311931 + -1.5 = -5.653311931 z = -5.653311931 Simplifying z = -5.653311931Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.653311931, -5.653311931}
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