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Simplifying 15z + -1z2 + -20 = 0 Reorder the terms: -20 + 15z + -1z2 = 0 Solving -20 + 15z + -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'. 20 + -15z + z2 = 0 Move the constant term to the right: Add '-20' to each side of the equation. 20 + -15z + -20 + z2 = 0 + -20 Reorder the terms: 20 + -20 + -15z + z2 = 0 + -20 Combine like terms: 20 + -20 = 0 0 + -15z + z2 = 0 + -20 -15z + z2 = 0 + -20 Combine like terms: 0 + -20 = -20 -15z + z2 = -20 The z term is -15z. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15z + 56.25 + z2 = -20 + 56.25 Reorder the terms: 56.25 + -15z + z2 = -20 + 56.25 Combine like terms: -20 + 56.25 = 36.25 56.25 + -15z + z2 = 36.25 Factor a perfect square on the left side: (z + -7.5)(z + -7.5) = 36.25 Calculate the square root of the right side: 6.020797289 Break this problem into two subproblems by setting (z + -7.5) equal to 6.020797289 and -6.020797289.Subproblem 1
z + -7.5 = 6.020797289 Simplifying z + -7.5 = 6.020797289 Reorder the terms: -7.5 + z = 6.020797289 Solving -7.5 + z = 6.020797289 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + z = 6.020797289 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + z = 6.020797289 + 7.5 z = 6.020797289 + 7.5 Combine like terms: 6.020797289 + 7.5 = 13.520797289 z = 13.520797289 Simplifying z = 13.520797289Subproblem 2
z + -7.5 = -6.020797289 Simplifying z + -7.5 = -6.020797289 Reorder the terms: -7.5 + z = -6.020797289 Solving -7.5 + z = -6.020797289 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + z = -6.020797289 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + z = -6.020797289 + 7.5 z = -6.020797289 + 7.5 Combine like terms: -6.020797289 + 7.5 = 1.479202711 z = 1.479202711 Simplifying z = 1.479202711Solution
The solution to the problem is based on the solutions from the subproblems. z = {13.520797289, 1.479202711}
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