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Simplifying 7(14x2 + 5x + -1) = 1 Reorder the terms: 7(-1 + 5x + 14x2) = 1 (-1 * 7 + 5x * 7 + 14x2 * 7) = 1 (-7 + 35x + 98x2) = 1 Solving -7 + 35x + 98x2 = 1 Solving for variable 'x'. Reorder the terms: -7 + -1 + 35x + 98x2 = 1 + -1 Combine like terms: -7 + -1 = -8 -8 + 35x + 98x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -8 + 35x + 98x2 = 0 Begin completing the square. Divide all terms by 98 the coefficient of the squared term: Divide each side by '98'. -0.08163265306 + 0.3571428571x + x2 = 0 Move the constant term to the right: Add '0.08163265306' to each side of the equation. -0.08163265306 + 0.3571428571x + 0.08163265306 + x2 = 0 + 0.08163265306 Reorder the terms: -0.08163265306 + 0.08163265306 + 0.3571428571x + x2 = 0 + 0.08163265306 Combine like terms: -0.08163265306 + 0.08163265306 = 0.00000000000 0.00000000000 + 0.3571428571x + x2 = 0 + 0.08163265306 0.3571428571x + x2 = 0 + 0.08163265306 Combine like terms: 0 + 0.08163265306 = 0.08163265306 0.3571428571x + x2 = 0.08163265306 The x term is 0.3571428571x. Take half its coefficient (0.1785714286). Square it (0.03188775511) and add it to both sides. Add '0.03188775511' to each side of the equation. 0.3571428571x + 0.03188775511 + x2 = 0.08163265306 + 0.03188775511 Reorder the terms: 0.03188775511 + 0.3571428571x + x2 = 0.08163265306 + 0.03188775511 Combine like terms: 0.08163265306 + 0.03188775511 = 0.11352040817 0.03188775511 + 0.3571428571x + x2 = 0.11352040817 Factor a perfect square on the left side: (x + 0.1785714286)(x + 0.1785714286) = 0.11352040817 Calculate the square root of the right side: 0.336927898 Break this problem into two subproblems by setting (x + 0.1785714286) equal to 0.336927898 and -0.336927898.Subproblem 1
x + 0.1785714286 = 0.336927898 Simplifying x + 0.1785714286 = 0.336927898 Reorder the terms: 0.1785714286 + x = 0.336927898 Solving 0.1785714286 + x = 0.336927898 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1785714286' to each side of the equation. 0.1785714286 + -0.1785714286 + x = 0.336927898 + -0.1785714286 Combine like terms: 0.1785714286 + -0.1785714286 = 0.0000000000 0.0000000000 + x = 0.336927898 + -0.1785714286 x = 0.336927898 + -0.1785714286 Combine like terms: 0.336927898 + -0.1785714286 = 0.1583564694 x = 0.1583564694 Simplifying x = 0.1583564694Subproblem 2
x + 0.1785714286 = -0.336927898 Simplifying x + 0.1785714286 = -0.336927898 Reorder the terms: 0.1785714286 + x = -0.336927898 Solving 0.1785714286 + x = -0.336927898 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1785714286' to each side of the equation. 0.1785714286 + -0.1785714286 + x = -0.336927898 + -0.1785714286 Combine like terms: 0.1785714286 + -0.1785714286 = 0.0000000000 0.0000000000 + x = -0.336927898 + -0.1785714286 x = -0.336927898 + -0.1785714286 Combine like terms: -0.336927898 + -0.1785714286 = -0.5154993266 x = -0.5154993266 Simplifying x = -0.5154993266Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.1583564694, -0.5154993266}
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