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Simplifying 14x + -1(3x + -2) + -1[5x + 2 + -1(x + -1)(x + -1)] = 0 Reorder the terms: 14x + -1(-2 + 3x) + -1[5x + 2 + -1(x + -1)(x + -1)] = 0 14x + (-2 * -1 + 3x * -1) + -1[5x + 2 + -1(x + -1)(x + -1)] = 0 14x + (2 + -3x) + -1[5x + 2 + -1(x + -1)(x + -1)] = 0 Reorder the terms: 14x + 2 + -3x + -1[5x + 2 + -1(-1 + x)(x + -1)] = 0 Reorder the terms: 14x + 2 + -3x + -1[5x + 2 + -1(-1 + x)(-1 + x)] = 0 Multiply (-1 + x) * (-1 + x) 14x + 2 + -3x + -1[5x + 2 + -1(-1(-1 + x) + x(-1 + x))] = 0 14x + 2 + -3x + -1[5x + 2 + -1((-1 * -1 + x * -1) + x(-1 + x))] = 0 14x + 2 + -3x + -1[5x + 2 + -1((1 + -1x) + x(-1 + x))] = 0 14x + 2 + -3x + -1[5x + 2 + -1(1 + -1x + (-1 * x + x * x))] = 0 14x + 2 + -3x + -1[5x + 2 + -1(1 + -1x + (-1x + x2))] = 0 Combine like terms: -1x + -1x = -2x 14x + 2 + -3x + -1[5x + 2 + -1(1 + -2x + x2)] = 0 14x + 2 + -3x + -1[5x + 2 + (1 * -1 + -2x * -1 + x2 * -1)] = 0 14x + 2 + -3x + -1[5x + 2 + (-1 + 2x + -1x2)] = 0 Reorder the terms: 14x + 2 + -3x + -1[2 + -1 + 5x + 2x + -1x2] = 0 Combine like terms: 2 + -1 = 1 14x + 2 + -3x + -1[1 + 5x + 2x + -1x2] = 0 Combine like terms: 5x + 2x = 7x 14x + 2 + -3x + -1[1 + 7x + -1x2] = 0 14x + 2 + -3x + [1 * -1 + 7x * -1 + -1x2 * -1] = 0 14x + 2 + -3x + [-1 + -7x + 1x2] = 0 Reorder the terms: 2 + -1 + 14x + -3x + -7x + 1x2 = 0 Combine like terms: 2 + -1 = 1 1 + 14x + -3x + -7x + 1x2 = 0 Combine like terms: 14x + -3x = 11x 1 + 11x + -7x + 1x2 = 0 Combine like terms: 11x + -7x = 4x 1 + 4x + 1x2 = 0 Solving 1 + 4x + 1x2 = 0 Solving for variable 'x'. Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + 4x + -1 + x2 = 0 + -1 Reorder the terms: 1 + -1 + 4x + x2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 4x + x2 = 0 + -1 4x + x2 = 0 + -1 Combine like terms: 0 + -1 = -1 4x + x2 = -1 The x term is 4x. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4x + 4 + x2 = -1 + 4 Reorder the terms: 4 + 4x + x2 = -1 + 4 Combine like terms: -1 + 4 = 3 4 + 4x + x2 = 3 Factor a perfect square on the left side: (x + 2)(x + 2) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (x + 2) equal to 1.732050808 and -1.732050808.Subproblem 1
x + 2 = 1.732050808 Simplifying x + 2 = 1.732050808 Reorder the terms: 2 + x = 1.732050808 Solving 2 + x = 1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + x = 1.732050808 + -2 Combine like terms: 2 + -2 = 0 0 + x = 1.732050808 + -2 x = 1.732050808 + -2 Combine like terms: 1.732050808 + -2 = -0.267949192 x = -0.267949192 Simplifying x = -0.267949192Subproblem 2
x + 2 = -1.732050808 Simplifying x + 2 = -1.732050808 Reorder the terms: 2 + x = -1.732050808 Solving 2 + x = -1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + x = -1.732050808 + -2 Combine like terms: 2 + -2 = 0 0 + x = -1.732050808 + -2 x = -1.732050808 + -2 Combine like terms: -1.732050808 + -2 = -3.732050808 x = -3.732050808 Simplifying x = -3.732050808Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.267949192, -3.732050808}
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