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Simplifying 6x2 + x = 82 Reorder the terms: x + 6x2 = 82 Solving x + 6x2 = 82 Solving for variable 'x'. Reorder the terms: -82 + x + 6x2 = 82 + -82 Combine like terms: 82 + -82 = 0 -82 + x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -13.66666667 + 0.1666666667x + x2 = 0 Move the constant term to the right: Add '13.66666667' to each side of the equation. -13.66666667 + 0.1666666667x + 13.66666667 + x2 = 0 + 13.66666667 Reorder the terms: -13.66666667 + 13.66666667 + 0.1666666667x + x2 = 0 + 13.66666667 Combine like terms: -13.66666667 + 13.66666667 = 0.00000000 0.00000000 + 0.1666666667x + x2 = 0 + 13.66666667 0.1666666667x + x2 = 0 + 13.66666667 Combine like terms: 0 + 13.66666667 = 13.66666667 0.1666666667x + x2 = 13.66666667 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. 0.1666666667x + 0.25 + x2 = 13.66666667 + 0.25 Reorder the terms: 0.25 + 0.1666666667x + x2 = 13.66666667 + 0.25 Combine like terms: 13.66666667 + 0.25 = 13.91666667 0.25 + 0.1666666667x + x2 = 13.91666667 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 13.91666667 Calculate the square root of the right side: 3.730504881 Break this problem into two subproblems by setting (x + 0.5) equal to 3.730504881 and -3.730504881.Subproblem 1
x + 0.5 = 3.730504881 Simplifying x + 0.5 = 3.730504881 Reorder the terms: 0.5 + x = 3.730504881 Solving 0.5 + x = 3.730504881 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 = 3.730504881 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 3.730504881 + -0.5 x = 3.730504881 + -0.5 Combine like terms: 3.730504881 + -0.5 = 3.230504881 x = 3.230504881 Simplifying x = 3.230504881Subproblem 2
x + 0.5 = -3.730504881 Simplifying x + 0.5 = -3.730504881 Reorder the terms: 0.5 + x = -3.730504881 Solving 0.5 + x = -3.730504881 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 = -3.730504881 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -3.730504881 + -0.5 x = -3.730504881 + -0.5 Combine like terms: -3.730504881 + -0.5 = -4.230504881 x = -4.230504881 Simplifying x = -4.230504881Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.230504881, -4.230504881}
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