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Simplifying z2 + -2.9z + 1 = 0 Reorder the terms: 1 + -2.9z + z2 = 0 Solving 1 + -2.9z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -2.9z + -1 + z2 = 0 + -1 Reorder the terms: 1 + -1 + -2.9z + z2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -2.9z + z2 = 0 + -1 -2.9z + z2 = 0 + -1 Combine like terms: 0 + -1 = -1 -2.9z + z2 = -1 The z term is -2.9z. Take half its coefficient (-1.45). Square it (2.1025) and add it to both sides. Add '2.1025' to each side of the equation. -2.9z + 2.1025 + z2 = -1 + 2.1025 Reorder the terms: 2.1025 + -2.9z + z2 = -1 + 2.1025 Combine like terms: -1 + 2.1025 = 1.1025 2.1025 + -2.9z + z2 = 1.1025 Factor a perfect square on the left side: (z + -1.45)(z + -1.45) = 1.1025 Calculate the square root of the right side: 1.05 Break this problem into two subproblems by setting (z + -1.45) equal to 1.05 and -1.05.Subproblem 1
z + -1.45 = 1.05 Simplifying z + -1.45 = 1.05 Reorder the terms: -1.45 + z = 1.05 Solving -1.45 + z = 1.05 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.45' to each side of the equation. -1.45 + 1.45 + z = 1.05 + 1.45 Combine like terms: -1.45 + 1.45 = 0.00 0.00 + z = 1.05 + 1.45 z = 1.05 + 1.45 Combine like terms: 1.05 + 1.45 = 2.5 z = 2.5 Simplifying z = 2.5Subproblem 2
z + -1.45 = -1.05 Simplifying z + -1.45 = -1.05 Reorder the terms: -1.45 + z = -1.05 Solving -1.45 + z = -1.05 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.45' to each side of the equation. -1.45 + 1.45 + z = -1.05 + 1.45 Combine like terms: -1.45 + 1.45 = 0.00 0.00 + z = -1.05 + 1.45 z = -1.05 + 1.45 Combine like terms: -1.05 + 1.45 = 0.4 z = 0.4 Simplifying z = 0.4Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.5, 0.4}
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