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Simplifying (3x + 2)(x + 2) = 100 Reorder the terms: (2 + 3x)(x + 2) = 100 Reorder the terms: (2 + 3x)(2 + x) = 100 Multiply (2 + 3x) * (2 + x) (2(2 + x) + 3x * (2 + x)) = 100 ((2 * 2 + x * 2) + 3x * (2 + x)) = 100 ((4 + 2x) + 3x * (2 + x)) = 100 (4 + 2x + (2 * 3x + x * 3x)) = 100 (4 + 2x + (6x + 3x2)) = 100 Combine like terms: 2x + 6x = 8x (4 + 8x + 3x2) = 100 Solving 4 + 8x + 3x2 = 100 Solving for variable 'x'. Reorder the terms: 4 + -100 + 8x + 3x2 = 100 + -100 Combine like terms: 4 + -100 = -96 -96 + 8x + 3x2 = 100 + -100 Combine like terms: 100 + -100 = 0 -96 + 8x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -32 + 2.666666667x + x2 = 0 Move the constant term to the right: Add '32' to each side of the equation. -32 + 2.666666667x + 32 + x2 = 0 + 32 Reorder the terms: -32 + 32 + 2.666666667x + x2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + 2.666666667x + x2 = 0 + 32 2.666666667x + x2 = 0 + 32 Combine like terms: 0 + 32 = 32 2.666666667x + x2 = 32 The x term is 2.666666667x. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667x + 1.777777780 + x2 = 32 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667x + x2 = 32 + 1.777777780 Combine like terms: 32 + 1.777777780 = 33.77777778 1.777777780 + 2.666666667x + x2 = 33.77777778 Factor a perfect square on the left side: (x + 1.333333334)(x + 1.333333334) = 33.77777778 Calculate the square root of the right side: 5.811865258 Break this problem into two subproblems by setting (x + 1.333333334) equal to 5.811865258 and -5.811865258.Subproblem 1
x + 1.333333334 = 5.811865258 Simplifying x + 1.333333334 = 5.811865258 Reorder the terms: 1.333333334 + x = 5.811865258 Solving 1.333333334 + x = 5.811865258 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = 5.811865258 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 5.811865258 + -1.333333334 x = 5.811865258 + -1.333333334 Combine like terms: 5.811865258 + -1.333333334 = 4.478531924 x = 4.478531924 Simplifying x = 4.478531924Subproblem 2
x + 1.333333334 = -5.811865258 Simplifying x + 1.333333334 = -5.811865258 Reorder the terms: 1.333333334 + x = -5.811865258 Solving 1.333333334 + x = -5.811865258 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = -5.811865258 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -5.811865258 + -1.333333334 x = -5.811865258 + -1.333333334 Combine like terms: -5.811865258 + -1.333333334 = -7.145198592 x = -7.145198592 Simplifying x = -7.145198592Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.478531924, -7.145198592}
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