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Simplifying 165 = (3x + 2) * x Reorder the terms: 165 = (2 + 3x) * x Reorder the terms for easier multiplication: 165 = x(2 + 3x) 165 = (2 * x + 3x * x) 165 = (2x + 3x2) Solving 165 = 2x + 3x2 Solving for variable 'x'. Reorder the terms: 165 + -2x + -3x2 = 2x + -2x + 3x2 + -3x2 Combine like terms: 2x + -2x = 0 165 + -2x + -3x2 = 0 + 3x2 + -3x2 165 + -2x + -3x2 = 3x2 + -3x2 Combine like terms: 3x2 + -3x2 = 0 165 + -2x + -3x2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -55 + 0.6666666667x + x2 = 0 Move the constant term to the right: Add '55' to each side of the equation. -55 + 0.6666666667x + 55 + x2 = 0 + 55 Reorder the terms: -55 + 55 + 0.6666666667x + x2 = 0 + 55 Combine like terms: -55 + 55 = 0 0 + 0.6666666667x + x2 = 0 + 55 0.6666666667x + x2 = 0 + 55 Combine like terms: 0 + 55 = 55 0.6666666667x + x2 = 55 The x term is 0.6666666667x. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667x + 0.1111111112 + x2 = 55 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667x + x2 = 55 + 0.1111111112 Combine like terms: 55 + 0.1111111112 = 55.1111111112 0.1111111112 + 0.6666666667x + x2 = 55.1111111112 Factor a perfect square on the left side: (x + 0.3333333334)(x + 0.3333333334) = 55.1111111112 Calculate the square root of the right side: 7.423685817 Break this problem into two subproblems by setting (x + 0.3333333334) equal to 7.423685817 and -7.423685817.Subproblem 1
x + 0.3333333334 = 7.423685817 Simplifying x + 0.3333333334 = 7.423685817 Reorder the terms: 0.3333333334 + x = 7.423685817 Solving 0.3333333334 + x = 7.423685817 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = 7.423685817 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 7.423685817 + -0.3333333334 x = 7.423685817 + -0.3333333334 Combine like terms: 7.423685817 + -0.3333333334 = 7.0903524836 x = 7.0903524836 Simplifying x = 7.0903524836Subproblem 2
x + 0.3333333334 = -7.423685817 Simplifying x + 0.3333333334 = -7.423685817 Reorder the terms: 0.3333333334 + x = -7.423685817 Solving 0.3333333334 + x = -7.423685817 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = -7.423685817 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -7.423685817 + -0.3333333334 x = -7.423685817 + -0.3333333334 Combine like terms: -7.423685817 + -0.3333333334 = -7.7570191504 x = -7.7570191504 Simplifying x = -7.7570191504Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.0903524836, -7.7570191504}
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