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Simplifying 8z2 + 7z + 1 = 0 Reorder the terms: 1 + 7z + 8z2 = 0 Solving 1 + 7z + 8z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.125 + 0.875z + z2 = 0 Move the constant term to the right: Add '-0.125' to each side of the equation. 0.125 + 0.875z + -0.125 + z2 = 0 + -0.125 Reorder the terms: 0.125 + -0.125 + 0.875z + z2 = 0 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + 0.875z + z2 = 0 + -0.125 0.875z + z2 = 0 + -0.125 Combine like terms: 0 + -0.125 = -0.125 0.875z + z2 = -0.125 The z term is 0.875z. Take half its coefficient (0.4375). Square it (0.19140625) and add it to both sides. Add '0.19140625' to each side of the equation. 0.875z + 0.19140625 + z2 = -0.125 + 0.19140625 Reorder the terms: 0.19140625 + 0.875z + z2 = -0.125 + 0.19140625 Combine like terms: -0.125 + 0.19140625 = 0.06640625 0.19140625 + 0.875z + z2 = 0.06640625 Factor a perfect square on the left side: (z + 0.4375)(z + 0.4375) = 0.06640625 Calculate the square root of the right side: 0.257694102 Break this problem into two subproblems by setting (z + 0.4375) equal to 0.257694102 and -0.257694102.Subproblem 1
z + 0.4375 = 0.257694102 Simplifying z + 0.4375 = 0.257694102 Reorder the terms: 0.4375 + z = 0.257694102 Solving 0.4375 + z = 0.257694102 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + z = 0.257694102 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + z = 0.257694102 + -0.4375 z = 0.257694102 + -0.4375 Combine like terms: 0.257694102 + -0.4375 = -0.179805898 z = -0.179805898 Simplifying z = -0.179805898Subproblem 2
z + 0.4375 = -0.257694102 Simplifying z + 0.4375 = -0.257694102 Reorder the terms: 0.4375 + z = -0.257694102 Solving 0.4375 + z = -0.257694102 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + z = -0.257694102 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + z = -0.257694102 + -0.4375 z = -0.257694102 + -0.4375 Combine like terms: -0.257694102 + -0.4375 = -0.695194102 z = -0.695194102 Simplifying z = -0.695194102Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.179805898, -0.695194102}
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