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Simplifying z2 + 0.84z + -0.4 = 0 Reorder the terms: -0.4 + 0.84z + z2 = 0 Solving -0.4 + 0.84z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '0.4' to each side of the equation. -0.4 + 0.84z + 0.4 + z2 = 0 + 0.4 Reorder the terms: -0.4 + 0.4 + 0.84z + z2 = 0 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + 0.84z + z2 = 0 + 0.4 0.84z + z2 = 0 + 0.4 Combine like terms: 0 + 0.4 = 0.4 0.84z + z2 = 0.4 The z term is 0.84z. Take half its coefficient (0.42). Square it (0.1764) and add it to both sides. Add '0.1764' to each side of the equation. 0.84z + 0.1764 + z2 = 0.4 + 0.1764 Reorder the terms: 0.1764 + 0.84z + z2 = 0.4 + 0.1764 Combine like terms: 0.4 + 0.1764 = 0.5764 0.1764 + 0.84z + z2 = 0.5764 Factor a perfect square on the left side: (z + 0.42)(z + 0.42) = 0.5764 Calculate the square root of the right side: 0.759210116 Break this problem into two subproblems by setting (z + 0.42) equal to 0.759210116 and -0.759210116.Subproblem 1
z + 0.42 = 0.759210116 Simplifying z + 0.42 = 0.759210116 Reorder the terms: 0.42 + z = 0.759210116 Solving 0.42 + z = 0.759210116 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.42' to each side of the equation. 0.42 + -0.42 + z = 0.759210116 + -0.42 Combine like terms: 0.42 + -0.42 = 0.00 0.00 + z = 0.759210116 + -0.42 z = 0.759210116 + -0.42 Combine like terms: 0.759210116 + -0.42 = 0.339210116 z = 0.339210116 Simplifying z = 0.339210116Subproblem 2
z + 0.42 = -0.759210116 Simplifying z + 0.42 = -0.759210116 Reorder the terms: 0.42 + z = -0.759210116 Solving 0.42 + z = -0.759210116 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.42' to each side of the equation. 0.42 + -0.42 + z = -0.759210116 + -0.42 Combine like terms: 0.42 + -0.42 = 0.00 0.00 + z = -0.759210116 + -0.42 z = -0.759210116 + -0.42 Combine like terms: -0.759210116 + -0.42 = -1.179210116 z = -1.179210116 Simplifying z = -1.179210116Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.339210116, -1.179210116}
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