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Simplifying 40.5x(2x + 8) = x + 729 Reorder the terms: 40.5x(8 + 2x) = x + 729 (8 * 40.5x + 2x * 40.5x) = x + 729 (324x + 81x2) = x + 729 Reorder the terms: 324x + 81x2 = 729 + x Solving 324x + 81x2 = 729 + x Solving for variable 'x'. Reorder the terms: -729 + 324x + -1x + 81x2 = 729 + x + -729 + -1x Combine like terms: 324x + -1x = 323x -729 + 323x + 81x2 = 729 + x + -729 + -1x Reorder the terms: -729 + 323x + 81x2 = 729 + -729 + x + -1x Combine like terms: 729 + -729 = 0 -729 + 323x + 81x2 = 0 + x + -1x -729 + 323x + 81x2 = x + -1x Combine like terms: x + -1x = 0 -729 + 323x + 81x2 = 0 Begin completing the square. Divide all terms by 81 the coefficient of the squared term: Divide each side by '81'. -9 + 3.987654321x + x2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + 3.987654321x + 9 + x2 = 0 + 9 Reorder the terms: -9 + 9 + 3.987654321x + x2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + 3.987654321x + x2 = 0 + 9 3.987654321x + x2 = 0 + 9 Combine like terms: 0 + 9 = 9 3.987654321x + x2 = 9 The x term is 3.987654321x. Take half its coefficient (1.993827161). Square it (3.975346748) and add it to both sides. Add '3.975346748' to each side of the equation. 3.987654321x + 3.975346748 + x2 = 9 + 3.975346748 Reorder the terms: 3.975346748 + 3.987654321x + x2 = 9 + 3.975346748 Combine like terms: 9 + 3.975346748 = 12.975346748 3.975346748 + 3.987654321x + x2 = 12.975346748 Factor a perfect square on the left side: (x + 1.993827161)(x + 1.993827161) = 12.975346748 Calculate the square root of the right side: 3.602130862 Break this problem into two subproblems by setting (x + 1.993827161) equal to 3.602130862 and -3.602130862.Subproblem 1
x + 1.993827161 = 3.602130862 Simplifying x + 1.993827161 = 3.602130862 Reorder the terms: 1.993827161 + x = 3.602130862 Solving 1.993827161 + x = 3.602130862 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.993827161' to each side of the equation. 1.993827161 + -1.993827161 + x = 3.602130862 + -1.993827161 Combine like terms: 1.993827161 + -1.993827161 = 0.000000000 0.000000000 + x = 3.602130862 + -1.993827161 x = 3.602130862 + -1.993827161 Combine like terms: 3.602130862 + -1.993827161 = 1.608303701 x = 1.608303701 Simplifying x = 1.608303701Subproblem 2
x + 1.993827161 = -3.602130862 Simplifying x + 1.993827161 = -3.602130862 Reorder the terms: 1.993827161 + x = -3.602130862 Solving 1.993827161 + x = -3.602130862 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.993827161' to each side of the equation. 1.993827161 + -1.993827161 + x = -3.602130862 + -1.993827161 Combine like terms: 1.993827161 + -1.993827161 = 0.000000000 0.000000000 + x = -3.602130862 + -1.993827161 x = -3.602130862 + -1.993827161 Combine like terms: -3.602130862 + -1.993827161 = -5.595958023 x = -5.595958023 Simplifying x = -5.595958023Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.608303701, -5.595958023}
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