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Simplifying 6x(x + 1) = 56 + -11x Reorder the terms: 6x(1 + x) = 56 + -11x (1 * 6x + x * 6x) = 56 + -11x (6x + 6x2) = 56 + -11x Solving 6x + 6x2 = 56 + -11x Solving for variable 'x'. Reorder the terms: -56 + 6x + 11x + 6x2 = 56 + -11x + -56 + 11x Combine like terms: 6x + 11x = 17x -56 + 17x + 6x2 = 56 + -11x + -56 + 11x Reorder the terms: -56 + 17x + 6x2 = 56 + -56 + -11x + 11x Combine like terms: 56 + -56 = 0 -56 + 17x + 6x2 = 0 + -11x + 11x -56 + 17x + 6x2 = -11x + 11x Combine like terms: -11x + 11x = 0 -56 + 17x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -9.333333333 + 2.833333333x + x2 = 0 Move the constant term to the right: Add '9.333333333' to each side of the equation. -9.333333333 + 2.833333333x + 9.333333333 + x2 = 0 + 9.333333333 Reorder the terms: -9.333333333 + 9.333333333 + 2.833333333x + x2 = 0 + 9.333333333 Combine like terms: -9.333333333 + 9.333333333 = 0.000000000 0.000000000 + 2.833333333x + x2 = 0 + 9.333333333 2.833333333x + x2 = 0 + 9.333333333 Combine like terms: 0 + 9.333333333 = 9.333333333 2.833333333x + x2 = 9.333333333 The x term is 2.833333333x. Take half its coefficient (1.416666667). Square it (2.006944445) and add it to both sides. Add '2.006944445' to each side of the equation. 2.833333333x + 2.006944445 + x2 = 9.333333333 + 2.006944445 Reorder the terms: 2.006944445 + 2.833333333x + x2 = 9.333333333 + 2.006944445 Combine like terms: 9.333333333 + 2.006944445 = 11.340277778 2.006944445 + 2.833333333x + x2 = 11.340277778 Factor a perfect square on the left side: (x + 1.416666667)(x + 1.416666667) = 11.340277778 Calculate the square root of the right side: 3.367532892 Break this problem into two subproblems by setting (x + 1.416666667) equal to 3.367532892 and -3.367532892.Subproblem 1
x + 1.416666667 = 3.367532892 Simplifying x + 1.416666667 = 3.367532892 Reorder the terms: 1.416666667 + x = 3.367532892 Solving 1.416666667 + x = 3.367532892 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.416666667' to each side of the equation. 1.416666667 + -1.416666667 + x = 3.367532892 + -1.416666667 Combine like terms: 1.416666667 + -1.416666667 = 0.000000000 0.000000000 + x = 3.367532892 + -1.416666667 x = 3.367532892 + -1.416666667 Combine like terms: 3.367532892 + -1.416666667 = 1.950866225 x = 1.950866225 Simplifying x = 1.950866225Subproblem 2
x + 1.416666667 = -3.367532892 Simplifying x + 1.416666667 = -3.367532892 Reorder the terms: 1.416666667 + x = -3.367532892 Solving 1.416666667 + x = -3.367532892 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.416666667' to each side of the equation. 1.416666667 + -1.416666667 + x = -3.367532892 + -1.416666667 Combine like terms: 1.416666667 + -1.416666667 = 0.000000000 0.000000000 + x = -3.367532892 + -1.416666667 x = -3.367532892 + -1.416666667 Combine like terms: -3.367532892 + -1.416666667 = -4.784199559 x = -4.784199559 Simplifying x = -4.784199559Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.950866225, -4.784199559}
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