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Simplifying 108 = [x(7x + 3)] + -60x Reorder the terms: 108 = [x(3 + 7x)] + -60x 108 = [(3 * x + 7x * x)] + -60x 108 = [(3x + 7x2)] + -60x 108 = [3x + 7x2] + -60x Remove brackets around [3x + 7x2] 108 = 3x + 7x2 + -60x Reorder the terms: 108 = 3x + -60x + 7x2 Combine like terms: 3x + -60x = -57x 108 = -57x + 7x2 Solving 108 = -57x + 7x2 Solving for variable 'x'. Reorder the terms: 108 + 57x + -7x2 = -57x + 57x + 7x2 + -7x2 Combine like terms: -57x + 57x = 0 108 + 57x + -7x2 = 0 + 7x2 + -7x2 108 + 57x + -7x2 = 7x2 + -7x2 Combine like terms: 7x2 + -7x2 = 0 108 + 57x + -7x2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -15.42857143 + -8.142857143x + x2 = 0 Move the constant term to the right: Add '15.42857143' to each side of the equation. -15.42857143 + -8.142857143x + 15.42857143 + x2 = 0 + 15.42857143 Reorder the terms: -15.42857143 + 15.42857143 + -8.142857143x + x2 = 0 + 15.42857143 Combine like terms: -15.42857143 + 15.42857143 = 0.00000000 0.00000000 + -8.142857143x + x2 = 0 + 15.42857143 -8.142857143x + x2 = 0 + 15.42857143 Combine like terms: 0 + 15.42857143 = 15.42857143 -8.142857143x + x2 = 15.42857143 The x term is -8.142857143x. Take half its coefficient (-4.071428572). Square it (16.57653062) and add it to both sides. Add '16.57653062' to each side of the equation. -8.142857143x + 16.57653062 + x2 = 15.42857143 + 16.57653062 Reorder the terms: 16.57653062 + -8.142857143x + x2 = 15.42857143 + 16.57653062 Combine like terms: 15.42857143 + 16.57653062 = 32.00510205 16.57653062 + -8.142857143x + x2 = 32.00510205 Factor a perfect square on the left side: (x + -4.071428572)(x + -4.071428572) = 32.00510205 Calculate the square root of the right side: 5.657305193 Break this problem into two subproblems by setting (x + -4.071428572) equal to 5.657305193 and -5.657305193.Subproblem 1
x + -4.071428572 = 5.657305193 Simplifying x + -4.071428572 = 5.657305193 Reorder the terms: -4.071428572 + x = 5.657305193 Solving -4.071428572 + x = 5.657305193 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.071428572' to each side of the equation. -4.071428572 + 4.071428572 + x = 5.657305193 + 4.071428572 Combine like terms: -4.071428572 + 4.071428572 = 0.000000000 0.000000000 + x = 5.657305193 + 4.071428572 x = 5.657305193 + 4.071428572 Combine like terms: 5.657305193 + 4.071428572 = 9.728733765 x = 9.728733765 Simplifying x = 9.728733765Subproblem 2
x + -4.071428572 = -5.657305193 Simplifying x + -4.071428572 = -5.657305193 Reorder the terms: -4.071428572 + x = -5.657305193 Solving -4.071428572 + x = -5.657305193 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.071428572' to each side of the equation. -4.071428572 + 4.071428572 + x = -5.657305193 + 4.071428572 Combine like terms: -4.071428572 + 4.071428572 = 0.000000000 0.000000000 + x = -5.657305193 + 4.071428572 x = -5.657305193 + 4.071428572 Combine like terms: -5.657305193 + 4.071428572 = -1.585876621 x = -1.585876621 Simplifying x = -1.585876621Solution
The solution to the problem is based on the solutions from the subproblems. x = {9.728733765, -1.585876621}
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