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Simplifying 50 + -1z2 + 10z = 0 Reorder the terms: 50 + 10z + -1z2 = 0 Solving 50 + 10z + -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'. -50 + -10z + z2 = 0 Move the constant term to the right: Add '50' to each side of the equation. -50 + -10z + 50 + z2 = 0 + 50 Reorder the terms: -50 + 50 + -10z + z2 = 0 + 50 Combine like terms: -50 + 50 = 0 0 + -10z + z2 = 0 + 50 -10z + z2 = 0 + 50 Combine like terms: 0 + 50 = 50 -10z + z2 = 50 The z term is -10z. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10z + 25 + z2 = 50 + 25 Reorder the terms: 25 + -10z + z2 = 50 + 25 Combine like terms: 50 + 25 = 75 25 + -10z + z2 = 75 Factor a perfect square on the left side: (z + -5)(z + -5) = 75 Calculate the square root of the right side: 8.660254038 Break this problem into two subproblems by setting (z + -5) equal to 8.660254038 and -8.660254038.Subproblem 1
z + -5 = 8.660254038 Simplifying z + -5 = 8.660254038 Reorder the terms: -5 + z = 8.660254038 Solving -5 + z = 8.660254038 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + z = 8.660254038 + 5 Combine like terms: -5 + 5 = 0 0 + z = 8.660254038 + 5 z = 8.660254038 + 5 Combine like terms: 8.660254038 + 5 = 13.660254038 z = 13.660254038 Simplifying z = 13.660254038Subproblem 2
z + -5 = -8.660254038 Simplifying z + -5 = -8.660254038 Reorder the terms: -5 + z = -8.660254038 Solving -5 + z = -8.660254038 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + z = -8.660254038 + 5 Combine like terms: -5 + 5 = 0 0 + z = -8.660254038 + 5 z = -8.660254038 + 5 Combine like terms: -8.660254038 + 5 = -3.660254038 z = -3.660254038 Simplifying z = -3.660254038Solution
The solution to the problem is based on the solutions from the subproblems. z = {13.660254038, -3.660254038}
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