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Simplifying 8x2 + 31x + 27 = 0 Reorder the terms: 27 + 31x + 8x2 = 0 Solving 27 + 31x + 8x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 3.375 + 3.875x + x2 = 0 Move the constant term to the right: Add '-3.375' to each side of the equation. 3.375 + 3.875x + -3.375 + x2 = 0 + -3.375 Reorder the terms: 3.375 + -3.375 + 3.875x + x2 = 0 + -3.375 Combine like terms: 3.375 + -3.375 = 0.000 0.000 + 3.875x + x2 = 0 + -3.375 3.875x + x2 = 0 + -3.375 Combine like terms: 0 + -3.375 = -3.375 3.875x + x2 = -3.375 The x term is 3.875x. Take half its coefficient (1.9375). Square it (3.75390625) and add it to both sides. Add '3.75390625' to each side of the equation. 3.875x + 3.75390625 + x2 = -3.375 + 3.75390625 Reorder the terms: 3.75390625 + 3.875x + x2 = -3.375 + 3.75390625 Combine like terms: -3.375 + 3.75390625 = 0.37890625 3.75390625 + 3.875x + x2 = 0.37890625 Factor a perfect square on the left side: (x + 1.9375)(x + 1.9375) = 0.37890625 Calculate the square root of the right side: 0.615553613 Break this problem into two subproblems by setting (x + 1.9375) equal to 0.615553613 and -0.615553613.Subproblem 1
x + 1.9375 = 0.615553613 Simplifying x + 1.9375 = 0.615553613 Reorder the terms: 1.9375 + x = 0.615553613 Solving 1.9375 + x = 0.615553613 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.9375' to each side of the equation. 1.9375 + -1.9375 + x = 0.615553613 + -1.9375 Combine like terms: 1.9375 + -1.9375 = 0.0000 0.0000 + x = 0.615553613 + -1.9375 x = 0.615553613 + -1.9375 Combine like terms: 0.615553613 + -1.9375 = -1.321946387 x = -1.321946387 Simplifying x = -1.321946387Subproblem 2
x + 1.9375 = -0.615553613 Simplifying x + 1.9375 = -0.615553613 Reorder the terms: 1.9375 + x = -0.615553613 Solving 1.9375 + x = -0.615553613 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.9375' to each side of the equation. 1.9375 + -1.9375 + x = -0.615553613 + -1.9375 Combine like terms: 1.9375 + -1.9375 = 0.0000 0.0000 + x = -0.615553613 + -1.9375 x = -0.615553613 + -1.9375 Combine like terms: -0.615553613 + -1.9375 = -2.553053613 x = -2.553053613 Simplifying x = -2.553053613Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.321946387, -2.553053613}
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