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Simplifying (2x2 + -4x + -61) + (7x2 + 7x) = 0 Reorder the terms: (-61 + -4x + 2x2) + (7x2 + 7x) = 0 Remove parenthesis around (-61 + -4x + 2x2) -61 + -4x + 2x2 + (7x2 + 7x) = 0 Reorder the terms: -61 + -4x + 2x2 + (7x + 7x2) = 0 Remove parenthesis around (7x + 7x2) -61 + -4x + 2x2 + 7x + 7x2 = 0 Reorder the terms: -61 + -4x + 7x + 2x2 + 7x2 = 0 Combine like terms: -4x + 7x = 3x -61 + 3x + 2x2 + 7x2 = 0 Combine like terms: 2x2 + 7x2 = 9x2 -61 + 3x + 9x2 = 0 Solving -61 + 3x + 9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -6.777777778 + 0.3333333333x + x2 = 0 Move the constant term to the right: Add '6.777777778' to each side of the equation. -6.777777778 + 0.3333333333x + 6.777777778 + x2 = 0 + 6.777777778 Reorder the terms: -6.777777778 + 6.777777778 + 0.3333333333x + x2 = 0 + 6.777777778 Combine like terms: -6.777777778 + 6.777777778 = 0.000000000 0.000000000 + 0.3333333333x + x2 = 0 + 6.777777778 0.3333333333x + x2 = 0 + 6.777777778 Combine like terms: 0 + 6.777777778 = 6.777777778 0.3333333333x + x2 = 6.777777778 The x term is 0.3333333333x. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333x + 0.02777777779 + x2 = 6.777777778 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333x + x2 = 6.777777778 + 0.02777777779 Combine like terms: 6.777777778 + 0.02777777779 = 6.80555555579 0.02777777779 + 0.3333333333x + x2 = 6.80555555579 Factor a perfect square on the left side: (x + 0.1666666667)(x + 0.1666666667) = 6.80555555579 Calculate the square root of the right side: 2.608745974 Break this problem into two subproblems by setting (x + 0.1666666667) equal to 2.608745974 and -2.608745974.Subproblem 1
x + 0.1666666667 = 2.608745974 Simplifying x + 0.1666666667 = 2.608745974 Reorder the terms: 0.1666666667 + x = 2.608745974 Solving 0.1666666667 + x = 2.608745974 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = 2.608745974 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = 2.608745974 + -0.1666666667 x = 2.608745974 + -0.1666666667 Combine like terms: 2.608745974 + -0.1666666667 = 2.4420793073 x = 2.4420793073 Simplifying x = 2.4420793073Subproblem 2
x + 0.1666666667 = -2.608745974 Simplifying x + 0.1666666667 = -2.608745974 Reorder the terms: 0.1666666667 + x = -2.608745974 Solving 0.1666666667 + x = -2.608745974 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = -2.608745974 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = -2.608745974 + -0.1666666667 x = -2.608745974 + -0.1666666667 Combine like terms: -2.608745974 + -0.1666666667 = -2.7754126407 x = -2.7754126407 Simplifying x = -2.7754126407Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.4420793073, -2.7754126407}
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