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Simplifying -1z2 + -21z + 102 = 0 Reorder the terms: 102 + -21z + -1z2 = 0 Solving 102 + -21z + -1z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -102 + 21z + z2 = 0 Move the constant term to the right: Add '102' to each side of the equation. -102 + 21z + 102 + z2 = 0 + 102 Reorder the terms: -102 + 102 + 21z + z2 = 0 + 102 Combine like terms: -102 + 102 = 0 0 + 21z + z2 = 0 + 102 21z + z2 = 0 + 102 Combine like terms: 0 + 102 = 102 21z + z2 = 102 The z term is 21z. Take half its coefficient (10.5). Square it (110.25) and add it to both sides. Add '110.25' to each side of the equation. 21z + 110.25 + z2 = 102 + 110.25 Reorder the terms: 110.25 + 21z + z2 = 102 + 110.25 Combine like terms: 102 + 110.25 = 212.25 110.25 + 21z + z2 = 212.25 Factor a perfect square on the left side: (z + 10.5)(z + 10.5) = 212.25 Calculate the square root of the right side: 14.568802284 Break this problem into two subproblems by setting (z + 10.5) equal to 14.568802284 and -14.568802284.Subproblem 1
z + 10.5 = 14.568802284 Simplifying z + 10.5 = 14.568802284 Reorder the terms: 10.5 + z = 14.568802284 Solving 10.5 + z = 14.568802284 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-10.5' to each side of the equation. 10.5 + -10.5 + z = 14.568802284 + -10.5 Combine like terms: 10.5 + -10.5 = 0.0 0.0 + z = 14.568802284 + -10.5 z = 14.568802284 + -10.5 Combine like terms: 14.568802284 + -10.5 = 4.068802284 z = 4.068802284 Simplifying z = 4.068802284Subproblem 2
z + 10.5 = -14.568802284 Simplifying z + 10.5 = -14.568802284 Reorder the terms: 10.5 + z = -14.568802284 Solving 10.5 + z = -14.568802284 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-10.5' to each side of the equation. 10.5 + -10.5 + z = -14.568802284 + -10.5 Combine like terms: 10.5 + -10.5 = 0.0 0.0 + z = -14.568802284 + -10.5 z = -14.568802284 + -10.5 Combine like terms: -14.568802284 + -10.5 = -25.068802284 z = -25.068802284 Simplifying z = -25.068802284Solution
The solution to the problem is based on the solutions from the subproblems. z = {4.068802284, -25.068802284}
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