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Simplifying (8x2 + -6x + -31) + (6x2 + 9x) = 0 Reorder the terms: (-31 + -6x + 8x2) + (6x2 + 9x) = 0 Remove parenthesis around (-31 + -6x + 8x2) -31 + -6x + 8x2 + (6x2 + 9x) = 0 Reorder the terms: -31 + -6x + 8x2 + (9x + 6x2) = 0 Remove parenthesis around (9x + 6x2) -31 + -6x + 8x2 + 9x + 6x2 = 0 Reorder the terms: -31 + -6x + 9x + 8x2 + 6x2 = 0 Combine like terms: -6x + 9x = 3x -31 + 3x + 8x2 + 6x2 = 0 Combine like terms: 8x2 + 6x2 = 14x2 -31 + 3x + 14x2 = 0 Solving -31 + 3x + 14x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -2.214285714 + 0.2142857143x + x2 = 0 Move the constant term to the right: Add '2.214285714' to each side of the equation. -2.214285714 + 0.2142857143x + 2.214285714 + x2 = 0 + 2.214285714 Reorder the terms: -2.214285714 + 2.214285714 + 0.2142857143x + x2 = 0 + 2.214285714 Combine like terms: -2.214285714 + 2.214285714 = 0.000000000 0.000000000 + 0.2142857143x + x2 = 0 + 2.214285714 0.2142857143x + x2 = 0 + 2.214285714 Combine like terms: 0 + 2.214285714 = 2.214285714 0.2142857143x + x2 = 2.214285714 The x term is 0.2142857143x. Take half its coefficient (0.1071428572). Square it (0.01147959185) and add it to both sides. Add '0.01147959185' to each side of the equation. 0.2142857143x + 0.01147959185 + x2 = 2.214285714 + 0.01147959185 Reorder the terms: 0.01147959185 + 0.2142857143x + x2 = 2.214285714 + 0.01147959185 Combine like terms: 2.214285714 + 0.01147959185 = 2.22576530585 0.01147959185 + 0.2142857143x + x2 = 2.22576530585 Factor a perfect square on the left side: (x + 0.1071428572)(x + 0.1071428572) = 2.22576530585 Calculate the square root of the right side: 1.491899898 Break this problem into two subproblems by setting (x + 0.1071428572) equal to 1.491899898 and -1.491899898.Subproblem 1
x + 0.1071428572 = 1.491899898 Simplifying x + 0.1071428572 = 1.491899898 Reorder the terms: 0.1071428572 + x = 1.491899898 Solving 0.1071428572 + x = 1.491899898 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1071428572' to each side of the equation. 0.1071428572 + -0.1071428572 + x = 1.491899898 + -0.1071428572 Combine like terms: 0.1071428572 + -0.1071428572 = 0.0000000000 0.0000000000 + x = 1.491899898 + -0.1071428572 x = 1.491899898 + -0.1071428572 Combine like terms: 1.491899898 + -0.1071428572 = 1.3847570408 x = 1.3847570408 Simplifying x = 1.3847570408Subproblem 2
x + 0.1071428572 = -1.491899898 Simplifying x + 0.1071428572 = -1.491899898 Reorder the terms: 0.1071428572 + x = -1.491899898 Solving 0.1071428572 + x = -1.491899898 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1071428572' to each side of the equation. 0.1071428572 + -0.1071428572 + x = -1.491899898 + -0.1071428572 Combine like terms: 0.1071428572 + -0.1071428572 = 0.0000000000 0.0000000000 + x = -1.491899898 + -0.1071428572 x = -1.491899898 + -0.1071428572 Combine like terms: -1.491899898 + -0.1071428572 = -1.5990427552 x = -1.5990427552 Simplifying x = -1.5990427552Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.3847570408, -1.5990427552}
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