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Simplifying (5x2 + 2x + 1) + -1(3x2 + -4x + -2) = 0 Reorder the terms: (1 + 2x + 5x2) + -1(3x2 + -4x + -2) = 0 Remove parenthesis around (1 + 2x + 5x2) 1 + 2x + 5x2 + -1(3x2 + -4x + -2) = 0 Reorder the terms: 1 + 2x + 5x2 + -1(-2 + -4x + 3x2) = 0 1 + 2x + 5x2 + (-2 * -1 + -4x * -1 + 3x2 * -1) = 0 1 + 2x + 5x2 + (2 + 4x + -3x2) = 0 Reorder the terms: 1 + 2 + 2x + 4x + 5x2 + -3x2 = 0 Combine like terms: 1 + 2 = 3 3 + 2x + 4x + 5x2 + -3x2 = 0 Combine like terms: 2x + 4x = 6x 3 + 6x + 5x2 + -3x2 = 0 Combine like terms: 5x2 + -3x2 = 2x2 3 + 6x + 2x2 = 0 Solving 3 + 6x + 2x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + 3x + x2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + 3x + -1.5 + x2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + 3x + x2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + 3x + x2 = 0 + -1.5 3x + x2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 3x + x2 = -1.5 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = -1.5 + 2.25 Reorder the terms: 2.25 + 3x + x2 = -1.5 + 2.25 Combine like terms: -1.5 + 2.25 = 0.75 2.25 + 3x + x2 = 0.75 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + 1.5) equal to 0.866025404 and -0.866025404.Subproblem 1
x + 1.5 = 0.866025404 Simplifying x + 1.5 = 0.866025404 Reorder the terms: 1.5 + x = 0.866025404 Solving 1.5 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 0.866025404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 0.866025404 + -1.5 x = 0.866025404 + -1.5 Combine like terms: 0.866025404 + -1.5 = -0.633974596 x = -0.633974596 Simplifying x = -0.633974596Subproblem 2
x + 1.5 = -0.866025404 Simplifying x + 1.5 = -0.866025404 Reorder the terms: 1.5 + x = -0.866025404 Solving 1.5 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -0.866025404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -0.866025404 + -1.5 x = -0.866025404 + -1.5 Combine like terms: -0.866025404 + -1.5 = -2.366025404 x = -2.366025404 Simplifying x = -2.366025404Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.633974596, -2.366025404}
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