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Simplifying 24z2 + 49z + -65 = 0 Reorder the terms: -65 + 49z + 24z2 = 0 Solving -65 + 49z + 24z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -2.708333333 + 2.041666667z + z2 = 0 Move the constant term to the right: Add '2.708333333' to each side of the equation. -2.708333333 + 2.041666667z + 2.708333333 + z2 = 0 + 2.708333333 Reorder the terms: -2.708333333 + 2.708333333 + 2.041666667z + z2 = 0 + 2.708333333 Combine like terms: -2.708333333 + 2.708333333 = 0.000000000 0.000000000 + 2.041666667z + z2 = 0 + 2.708333333 2.041666667z + z2 = 0 + 2.708333333 Combine like terms: 0 + 2.708333333 = 2.708333333 2.041666667z + z2 = 2.708333333 The z term is 2.041666667z. Take half its coefficient (1.020833334). Square it (1.042100696) and add it to both sides. Add '1.042100696' to each side of the equation. 2.041666667z + 1.042100696 + z2 = 2.708333333 + 1.042100696 Reorder the terms: 1.042100696 + 2.041666667z + z2 = 2.708333333 + 1.042100696 Combine like terms: 2.708333333 + 1.042100696 = 3.750434029 1.042100696 + 2.041666667z + z2 = 3.750434029 Factor a perfect square on the left side: (z + 1.020833334)(z + 1.020833334) = 3.750434029 Calculate the square root of the right side: 1.936603736 Break this problem into two subproblems by setting (z + 1.020833334) equal to 1.936603736 and -1.936603736.Subproblem 1
z + 1.020833334 = 1.936603736 Simplifying z + 1.020833334 = 1.936603736 Reorder the terms: 1.020833334 + z = 1.936603736 Solving 1.020833334 + z = 1.936603736 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.020833334' to each side of the equation. 1.020833334 + -1.020833334 + z = 1.936603736 + -1.020833334 Combine like terms: 1.020833334 + -1.020833334 = 0.000000000 0.000000000 + z = 1.936603736 + -1.020833334 z = 1.936603736 + -1.020833334 Combine like terms: 1.936603736 + -1.020833334 = 0.915770402 z = 0.915770402 Simplifying z = 0.915770402Subproblem 2
z + 1.020833334 = -1.936603736 Simplifying z + 1.020833334 = -1.936603736 Reorder the terms: 1.020833334 + z = -1.936603736 Solving 1.020833334 + z = -1.936603736 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.020833334' to each side of the equation. 1.020833334 + -1.020833334 + z = -1.936603736 + -1.020833334 Combine like terms: 1.020833334 + -1.020833334 = 0.000000000 0.000000000 + z = -1.936603736 + -1.020833334 z = -1.936603736 + -1.020833334 Combine like terms: -1.936603736 + -1.020833334 = -2.95743707 z = -2.95743707 Simplifying z = -2.95743707Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.915770402, -2.95743707}
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