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Simplifying 8z + -10 = z2 + -7z + 3 Reorder the terms: -10 + 8z = z2 + -7z + 3 Reorder the terms: -10 + 8z = 3 + -7z + z2 Solving -10 + 8z = 3 + -7z + z2 Solving for variable 'z'. Reorder the terms: -10 + -3 + 8z + 7z + -1z2 = 3 + -7z + z2 + -3 + 7z + -1z2 Combine like terms: -10 + -3 = -13 -13 + 8z + 7z + -1z2 = 3 + -7z + z2 + -3 + 7z + -1z2 Combine like terms: 8z + 7z = 15z -13 + 15z + -1z2 = 3 + -7z + z2 + -3 + 7z + -1z2 Reorder the terms: -13 + 15z + -1z2 = 3 + -3 + -7z + 7z + z2 + -1z2 Combine like terms: 3 + -3 = 0 -13 + 15z + -1z2 = 0 + -7z + 7z + z2 + -1z2 -13 + 15z + -1z2 = -7z + 7z + z2 + -1z2 Combine like terms: -7z + 7z = 0 -13 + 15z + -1z2 = 0 + z2 + -1z2 -13 + 15z + -1z2 = z2 + -1z2 Combine like terms: z2 + -1z2 = 0 -13 + 15z + -1z2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 13 + -15z + z2 = 0 Move the constant term to the right: Add '-13' to each side of the equation. 13 + -15z + -13 + z2 = 0 + -13 Reorder the terms: 13 + -13 + -15z + z2 = 0 + -13 Combine like terms: 13 + -13 = 0 0 + -15z + z2 = 0 + -13 -15z + z2 = 0 + -13 Combine like terms: 0 + -13 = -13 -15z + z2 = -13 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 = -13 + 56.25 Reorder the terms: 56.25 + -15z + z2 = -13 + 56.25 Combine like terms: -13 + 56.25 = 43.25 56.25 + -15z + z2 = 43.25 Factor a perfect square on the left side: (z + -7.5)(z + -7.5) = 43.25 Calculate the square root of the right side: 6.576473219 Break this problem into two subproblems by setting (z + -7.5) equal to 6.576473219 and -6.576473219.Subproblem 1
z + -7.5 = 6.576473219 Simplifying z + -7.5 = 6.576473219 Reorder the terms: -7.5 + z = 6.576473219 Solving -7.5 + z = 6.576473219 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 = 6.576473219 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + z = 6.576473219 + 7.5 z = 6.576473219 + 7.5 Combine like terms: 6.576473219 + 7.5 = 14.076473219 z = 14.076473219 Simplifying z = 14.076473219Subproblem 2
z + -7.5 = -6.576473219 Simplifying z + -7.5 = -6.576473219 Reorder the terms: -7.5 + z = -6.576473219 Solving -7.5 + z = -6.576473219 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 = -6.576473219 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + z = -6.576473219 + 7.5 z = -6.576473219 + 7.5 Combine like terms: -6.576473219 + 7.5 = 0.923526781 z = 0.923526781 Simplifying z = 0.923526781Solution
The solution to the problem is based on the solutions from the subproblems. z = {14.076473219, 0.923526781}
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