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Simplifying (5x + 1)(3x + -1) + -1(4x + -1)(x + 2) = 0 Reorder the terms: (1 + 5x)(3x + -1) + -1(4x + -1)(x + 2) = 0 Reorder the terms: (1 + 5x)(-1 + 3x) + -1(4x + -1)(x + 2) = 0 Multiply (1 + 5x) * (-1 + 3x) (1(-1 + 3x) + 5x * (-1 + 3x)) + -1(4x + -1)(x + 2) = 0 ((-1 * 1 + 3x * 1) + 5x * (-1 + 3x)) + -1(4x + -1)(x + 2) = 0 ((-1 + 3x) + 5x * (-1 + 3x)) + -1(4x + -1)(x + 2) = 0 (-1 + 3x + (-1 * 5x + 3x * 5x)) + -1(4x + -1)(x + 2) = 0 (-1 + 3x + (-5x + 15x2)) + -1(4x + -1)(x + 2) = 0 Combine like terms: 3x + -5x = -2x (-1 + -2x + 15x2) + -1(4x + -1)(x + 2) = 0 Reorder the terms: -1 + -2x + 15x2 + -1(-1 + 4x)(x + 2) = 0 Reorder the terms: -1 + -2x + 15x2 + -1(-1 + 4x)(2 + x) = 0 Multiply (-1 + 4x) * (2 + x) -1 + -2x + 15x2 + -1(-1(2 + x) + 4x * (2 + x)) = 0 -1 + -2x + 15x2 + -1((2 * -1 + x * -1) + 4x * (2 + x)) = 0 -1 + -2x + 15x2 + -1((-2 + -1x) + 4x * (2 + x)) = 0 -1 + -2x + 15x2 + -1(-2 + -1x + (2 * 4x + x * 4x)) = 0 -1 + -2x + 15x2 + -1(-2 + -1x + (8x + 4x2)) = 0 Combine like terms: -1x + 8x = 7x -1 + -2x + 15x2 + -1(-2 + 7x + 4x2) = 0 -1 + -2x + 15x2 + (-2 * -1 + 7x * -1 + 4x2 * -1) = 0 -1 + -2x + 15x2 + (2 + -7x + -4x2) = 0 Reorder the terms: -1 + 2 + -2x + -7x + 15x2 + -4x2 = 0 Combine like terms: -1 + 2 = 1 1 + -2x + -7x + 15x2 + -4x2 = 0 Combine like terms: -2x + -7x = -9x 1 + -9x + 15x2 + -4x2 = 0 Combine like terms: 15x2 + -4x2 = 11x2 1 + -9x + 11x2 = 0 Solving 1 + -9x + 11x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. 0.09090909091 + -0.8181818182x + x2 = 0 Move the constant term to the right: Add '-0.09090909091' to each side of the equation. 0.09090909091 + -0.8181818182x + -0.09090909091 + x2 = 0 + -0.09090909091 Reorder the terms: 0.09090909091 + -0.09090909091 + -0.8181818182x + x2 = 0 + -0.09090909091 Combine like terms: 0.09090909091 + -0.09090909091 = 0.00000000000 0.00000000000 + -0.8181818182x + x2 = 0 + -0.09090909091 -0.8181818182x + x2 = 0 + -0.09090909091 Combine like terms: 0 + -0.09090909091 = -0.09090909091 -0.8181818182x + x2 = -0.09090909091 The x term is -0.8181818182x. Take half its coefficient (-0.4090909091). Square it (0.1673553719) and add it to both sides. Add '0.1673553719' to each side of the equation. -0.8181818182x + 0.1673553719 + x2 = -0.09090909091 + 0.1673553719 Reorder the terms: 0.1673553719 + -0.8181818182x + x2 = -0.09090909091 + 0.1673553719 Combine like terms: -0.09090909091 + 0.1673553719 = 0.07644628099 0.1673553719 + -0.8181818182x + x2 = 0.07644628099 Factor a perfect square on the left side: (x + -0.4090909091)(x + -0.4090909091) = 0.07644628099 Calculate the square root of the right side: 0.276489206 Break this problem into two subproblems by setting (x + -0.4090909091) equal to 0.276489206 and -0.276489206.Subproblem 1
x + -0.4090909091 = 0.276489206 Simplifying x + -0.4090909091 = 0.276489206 Reorder the terms: -0.4090909091 + x = 0.276489206 Solving -0.4090909091 + x = 0.276489206 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4090909091' to each side of the equation. -0.4090909091 + 0.4090909091 + x = 0.276489206 + 0.4090909091 Combine like terms: -0.4090909091 + 0.4090909091 = 0.0000000000 0.0000000000 + x = 0.276489206 + 0.4090909091 x = 0.276489206 + 0.4090909091 Combine like terms: 0.276489206 + 0.4090909091 = 0.6855801151 x = 0.6855801151 Simplifying x = 0.6855801151Subproblem 2
x + -0.4090909091 = -0.276489206 Simplifying x + -0.4090909091 = -0.276489206 Reorder the terms: -0.4090909091 + x = -0.276489206 Solving -0.4090909091 + x = -0.276489206 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4090909091' to each side of the equation. -0.4090909091 + 0.4090909091 + x = -0.276489206 + 0.4090909091 Combine like terms: -0.4090909091 + 0.4090909091 = 0.0000000000 0.0000000000 + x = -0.276489206 + 0.4090909091 x = -0.276489206 + 0.4090909091 Combine like terms: -0.276489206 + 0.4090909091 = 0.1326017031 x = 0.1326017031 Simplifying x = 0.1326017031Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.6855801151, 0.1326017031}
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