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Simplifying (2x2 + 3x + -2) + (4x + 3) + (4x2 + 6x + -7) = 0 Reorder the terms: (-2 + 3x + 2x2) + (4x + 3) + (4x2 + 6x + -7) = 0 Remove parenthesis around (-2 + 3x + 2x2) -2 + 3x + 2x2 + (4x + 3) + (4x2 + 6x + -7) = 0 Reorder the terms: -2 + 3x + 2x2 + (3 + 4x) + (4x2 + 6x + -7) = 0 Remove parenthesis around (3 + 4x) -2 + 3x + 2x2 + 3 + 4x + (4x2 + 6x + -7) = 0 Reorder the terms: -2 + 3x + 2x2 + 3 + 4x + (-7 + 6x + 4x2) = 0 Remove parenthesis around (-7 + 6x + 4x2) -2 + 3x + 2x2 + 3 + 4x + -7 + 6x + 4x2 = 0 Reorder the terms: -2 + 3 + -7 + 3x + 4x + 6x + 2x2 + 4x2 = 0 Combine like terms: -2 + 3 = 1 1 + -7 + 3x + 4x + 6x + 2x2 + 4x2 = 0 Combine like terms: 1 + -7 = -6 -6 + 3x + 4x + 6x + 2x2 + 4x2 = 0 Combine like terms: 3x + 4x = 7x -6 + 7x + 6x + 2x2 + 4x2 = 0 Combine like terms: 7x + 6x = 13x -6 + 13x + 2x2 + 4x2 = 0 Combine like terms: 2x2 + 4x2 = 6x2 -6 + 13x + 6x2 = 0 Solving -6 + 13x + 6x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1 + 2.166666667x + x2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 2.166666667x + 1 + x2 = 0 + 1 Reorder the terms: -1 + 1 + 2.166666667x + x2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 2.166666667x + x2 = 0 + 1 2.166666667x + x2 = 0 + 1 Combine like terms: 0 + 1 = 1 2.166666667x + x2 = 1 The x term is 2.166666667x. Take half its coefficient (1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. 2.166666667x + 1.173611113 + x2 = 1 + 1.173611113 Reorder the terms: 1.173611113 + 2.166666667x + x2 = 1 + 1.173611113 Combine like terms: 1 + 1.173611113 = 2.173611113 1.173611113 + 2.166666667x + x2 = 2.173611113 Factor a perfect square on the left side: (x + 1.083333334)(x + 1.083333334) = 2.173611113 Calculate the square root of the right side: 1.474317168 Break this problem into two subproblems by setting (x + 1.083333334) equal to 1.474317168 and -1.474317168.Subproblem 1
x + 1.083333334 = 1.474317168 Simplifying x + 1.083333334 = 1.474317168 Reorder the terms: 1.083333334 + x = 1.474317168 Solving 1.083333334 + x = 1.474317168 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + x = 1.474317168 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + x = 1.474317168 + -1.083333334 x = 1.474317168 + -1.083333334 Combine like terms: 1.474317168 + -1.083333334 = 0.390983834 x = 0.390983834 Simplifying x = 0.390983834Subproblem 2
x + 1.083333334 = -1.474317168 Simplifying x + 1.083333334 = -1.474317168 Reorder the terms: 1.083333334 + x = -1.474317168 Solving 1.083333334 + x = -1.474317168 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + x = -1.474317168 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + x = -1.474317168 + -1.083333334 x = -1.474317168 + -1.083333334 Combine like terms: -1.474317168 + -1.083333334 = -2.557650502 x = -2.557650502 Simplifying x = -2.557650502Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.390983834, -2.557650502}
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