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Simplifying (2x + -5) * 3(4x + 7) = 2 Reorder the terms: (-5 + 2x) * 3(4x + 7) = 2 Reorder the terms: (-5 + 2x) * 3(7 + 4x) = 2 Reorder the terms for easier multiplication: 3(-5 + 2x)(7 + 4x) = 2 Multiply (-5 + 2x) * (7 + 4x) 3(-5(7 + 4x) + 2x * (7 + 4x)) = 2 3((7 * -5 + 4x * -5) + 2x * (7 + 4x)) = 2 3((-35 + -20x) + 2x * (7 + 4x)) = 2 3(-35 + -20x + (7 * 2x + 4x * 2x)) = 2 3(-35 + -20x + (14x + 8x2)) = 2 Combine like terms: -20x + 14x = -6x 3(-35 + -6x + 8x2) = 2 (-35 * 3 + -6x * 3 + 8x2 * 3) = 2 (-105 + -18x + 24x2) = 2 Solving -105 + -18x + 24x2 = 2 Solving for variable 'x'. Reorder the terms: -105 + -2 + -18x + 24x2 = 2 + -2 Combine like terms: -105 + -2 = -107 -107 + -18x + 24x2 = 2 + -2 Combine like terms: 2 + -2 = 0 -107 + -18x + 24x2 = 0 Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -4.458333333 + -0.75x + x2 = 0 Move the constant term to the right: Add '4.458333333' to each side of the equation. -4.458333333 + -0.75x + 4.458333333 + x2 = 0 + 4.458333333 Reorder the terms: -4.458333333 + 4.458333333 + -0.75x + x2 = 0 + 4.458333333 Combine like terms: -4.458333333 + 4.458333333 = 0.000000000 0.000000000 + -0.75x + x2 = 0 + 4.458333333 -0.75x + x2 = 0 + 4.458333333 Combine like terms: 0 + 4.458333333 = 4.458333333 -0.75x + x2 = 4.458333333 The x term is -0.75x. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75x + 0.140625 + x2 = 4.458333333 + 0.140625 Reorder the terms: 0.140625 + -0.75x + x2 = 4.458333333 + 0.140625 Combine like terms: 4.458333333 + 0.140625 = 4.598958333 0.140625 + -0.75x + x2 = 4.598958333 Factor a perfect square on the left side: (x + -0.375)(x + -0.375) = 4.598958333 Calculate the square root of the right side: 2.144518205 Break this problem into two subproblems by setting (x + -0.375) equal to 2.144518205 and -2.144518205.Subproblem 1
x + -0.375 = 2.144518205 Simplifying x + -0.375 = 2.144518205 Reorder the terms: -0.375 + x = 2.144518205 Solving -0.375 + x = 2.144518205 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = 2.144518205 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = 2.144518205 + 0.375 x = 2.144518205 + 0.375 Combine like terms: 2.144518205 + 0.375 = 2.519518205 x = 2.519518205 Simplifying x = 2.519518205Subproblem 2
x + -0.375 = -2.144518205 Simplifying x + -0.375 = -2.144518205 Reorder the terms: -0.375 + x = -2.144518205 Solving -0.375 + x = -2.144518205 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = -2.144518205 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = -2.144518205 + 0.375 x = -2.144518205 + 0.375 Combine like terms: -2.144518205 + 0.375 = -1.769518205 x = -1.769518205 Simplifying x = -1.769518205Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.519518205, -1.769518205}
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