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Simplifying 3x(x + 12) + 2x(2x + 4) = 360 Reorder the terms: 3x(12 + x) + 2x(2x + 4) = 360 (12 * 3x + x * 3x) + 2x(2x + 4) = 360 (36x + 3x2) + 2x(2x + 4) = 360 Reorder the terms: 36x + 3x2 + 2x(4 + 2x) = 360 36x + 3x2 + (4 * 2x + 2x * 2x) = 360 36x + 3x2 + (8x + 4x2) = 360 Reorder the terms: 36x + 8x + 3x2 + 4x2 = 360 Combine like terms: 36x + 8x = 44x 44x + 3x2 + 4x2 = 360 Combine like terms: 3x2 + 4x2 = 7x2 44x + 7x2 = 360 Solving 44x + 7x2 = 360 Solving for variable 'x'. Reorder the terms: -360 + 44x + 7x2 = 360 + -360 Combine like terms: 360 + -360 = 0 -360 + 44x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -51.42857143 + 6.285714286x + x2 = 0 Move the constant term to the right: Add '51.42857143' to each side of the equation. -51.42857143 + 6.285714286x + 51.42857143 + x2 = 0 + 51.42857143 Reorder the terms: -51.42857143 + 51.42857143 + 6.285714286x + x2 = 0 + 51.42857143 Combine like terms: -51.42857143 + 51.42857143 = 0.00000000 0.00000000 + 6.285714286x + x2 = 0 + 51.42857143 6.285714286x + x2 = 0 + 51.42857143 Combine like terms: 0 + 51.42857143 = 51.42857143 6.285714286x + x2 = 51.42857143 The x term is 6.285714286x. Take half its coefficient (3.142857143). Square it (9.877551021) and add it to both sides. Add '9.877551021' to each side of the equation. 6.285714286x + 9.877551021 + x2 = 51.42857143 + 9.877551021 Reorder the terms: 9.877551021 + 6.285714286x + x2 = 51.42857143 + 9.877551021 Combine like terms: 51.42857143 + 9.877551021 = 61.306122451 9.877551021 + 6.285714286x + x2 = 61.306122451 Factor a perfect square on the left side: (x + 3.142857143)(x + 3.142857143) = 61.306122451 Calculate the square root of the right side: 7.829822632 Break this problem into two subproblems by setting (x + 3.142857143) equal to 7.829822632 and -7.829822632.Subproblem 1
x + 3.142857143 = 7.829822632 Simplifying x + 3.142857143 = 7.829822632 Reorder the terms: 3.142857143 + x = 7.829822632 Solving 3.142857143 + x = 7.829822632 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.142857143' to each side of the equation. 3.142857143 + -3.142857143 + x = 7.829822632 + -3.142857143 Combine like terms: 3.142857143 + -3.142857143 = 0.000000000 0.000000000 + x = 7.829822632 + -3.142857143 x = 7.829822632 + -3.142857143 Combine like terms: 7.829822632 + -3.142857143 = 4.686965489 x = 4.686965489 Simplifying x = 4.686965489Subproblem 2
x + 3.142857143 = -7.829822632 Simplifying x + 3.142857143 = -7.829822632 Reorder the terms: 3.142857143 + x = -7.829822632 Solving 3.142857143 + x = -7.829822632 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.142857143' to each side of the equation. 3.142857143 + -3.142857143 + x = -7.829822632 + -3.142857143 Combine like terms: 3.142857143 + -3.142857143 = 0.000000000 0.000000000 + x = -7.829822632 + -3.142857143 x = -7.829822632 + -3.142857143 Combine like terms: -7.829822632 + -3.142857143 = -10.972679775 x = -10.972679775 Simplifying x = -10.972679775Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.686965489, -10.972679775}
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