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Simplifying 7z2 + -41z + 48 = 0 Reorder the terms: 48 + -41z + 7z2 = 0 Solving 48 + -41z + 7z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 6.857142857 + -5.857142857z + z2 = 0 Move the constant term to the right: Add '-6.857142857' to each side of the equation. 6.857142857 + -5.857142857z + -6.857142857 + z2 = 0 + -6.857142857 Reorder the terms: 6.857142857 + -6.857142857 + -5.857142857z + z2 = 0 + -6.857142857 Combine like terms: 6.857142857 + -6.857142857 = 0.000000000 0.000000000 + -5.857142857z + z2 = 0 + -6.857142857 -5.857142857z + z2 = 0 + -6.857142857 Combine like terms: 0 + -6.857142857 = -6.857142857 -5.857142857z + z2 = -6.857142857 The z term is -5.857142857z. Take half its coefficient (-2.928571429). Square it (8.576530615) and add it to both sides. Add '8.576530615' to each side of the equation. -5.857142857z + 8.576530615 + z2 = -6.857142857 + 8.576530615 Reorder the terms: 8.576530615 + -5.857142857z + z2 = -6.857142857 + 8.576530615 Combine like terms: -6.857142857 + 8.576530615 = 1.719387758 8.576530615 + -5.857142857z + z2 = 1.719387758 Factor a perfect square on the left side: (z + -2.928571429)(z + -2.928571429) = 1.719387758 Calculate the square root of the right side: 1.311254269 Break this problem into two subproblems by setting (z + -2.928571429) equal to 1.311254269 and -1.311254269.Subproblem 1
z + -2.928571429 = 1.311254269 Simplifying z + -2.928571429 = 1.311254269 Reorder the terms: -2.928571429 + z = 1.311254269 Solving -2.928571429 + z = 1.311254269 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.928571429' to each side of the equation. -2.928571429 + 2.928571429 + z = 1.311254269 + 2.928571429 Combine like terms: -2.928571429 + 2.928571429 = 0.000000000 0.000000000 + z = 1.311254269 + 2.928571429 z = 1.311254269 + 2.928571429 Combine like terms: 1.311254269 + 2.928571429 = 4.239825698 z = 4.239825698 Simplifying z = 4.239825698Subproblem 2
z + -2.928571429 = -1.311254269 Simplifying z + -2.928571429 = -1.311254269 Reorder the terms: -2.928571429 + z = -1.311254269 Solving -2.928571429 + z = -1.311254269 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.928571429' to each side of the equation. -2.928571429 + 2.928571429 + z = -1.311254269 + 2.928571429 Combine like terms: -2.928571429 + 2.928571429 = 0.000000000 0.000000000 + z = -1.311254269 + 2.928571429 z = -1.311254269 + 2.928571429 Combine like terms: -1.311254269 + 2.928571429 = 1.61731716 z = 1.61731716 Simplifying z = 1.61731716Solution
The solution to the problem is based on the solutions from the subproblems. z = {4.239825698, 1.61731716}
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