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Simplifying (3x2 + -4x + 6) + -1(-2x + 4) + (5x + -3) = 0 Reorder the terms: (6 + -4x + 3x2) + -1(-2x + 4) + (5x + -3) = 0 Remove parenthesis around (6 + -4x + 3x2) 6 + -4x + 3x2 + -1(-2x + 4) + (5x + -3) = 0 Reorder the terms: 6 + -4x + 3x2 + -1(4 + -2x) + (5x + -3) = 0 6 + -4x + 3x2 + (4 * -1 + -2x * -1) + (5x + -3) = 0 6 + -4x + 3x2 + (-4 + 2x) + (5x + -3) = 0 Reorder the terms: 6 + -4x + 3x2 + -4 + 2x + (-3 + 5x) = 0 Remove parenthesis around (-3 + 5x) 6 + -4x + 3x2 + -4 + 2x + -3 + 5x = 0 Reorder the terms: 6 + -4 + -3 + -4x + 2x + 5x + 3x2 = 0 Combine like terms: 6 + -4 = 2 2 + -3 + -4x + 2x + 5x + 3x2 = 0 Combine like terms: 2 + -3 = -1 -1 + -4x + 2x + 5x + 3x2 = 0 Combine like terms: -4x + 2x = -2x -1 + -2x + 5x + 3x2 = 0 Combine like terms: -2x + 5x = 3x -1 + 3x + 3x2 = 0 Solving -1 + 3x + 3x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.3333333333 + x + x2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + x + 0.3333333333 + x2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + x + x2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + x + x2 = 0 + 0.3333333333 x + x2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 x + x2 = 0.3333333333 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + x2 = 0.3333333333 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 0.3333333333 + 0.25 Combine like terms: 0.3333333333 + 0.25 = 0.5833333333 1.25 + x2 = 0.5833333333 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.5833333333 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (x + 0.5) equal to 0.763762616 and -0.763762616.Subproblem 1
x + 0.5 = 0.763762616 Simplifying x + 0.5 = 0.763762616 Reorder the terms: 0.5 + x = 0.763762616 Solving 0.5 + x = 0.763762616 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.763762616 + -0.5 x = 0.763762616 + -0.5 Combine like terms: 0.763762616 + -0.5 = 0.263762616 x = 0.263762616 Simplifying x = 0.263762616Subproblem 2
x + 0.5 = -0.763762616 Simplifying x + 0.5 = -0.763762616 Reorder the terms: 0.5 + x = -0.763762616 Solving 0.5 + x = -0.763762616 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.763762616 + -0.5 x = -0.763762616 + -0.5 Combine like terms: -0.763762616 + -0.5 = -1.263762616 x = -1.263762616 Simplifying x = -1.263762616Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.263762616, -1.263762616}
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