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Simplifying (3x + -7) * 2 + 5(2x + 1)(x + -2) = -1x2 + -1[-1(3x + 1)] Reorder the terms: (-7 + 3x) * 2 + 5(2x + 1)(x + -2) = -1x2 + -1[-1(3x + 1)] Reorder the terms for easier multiplication: 2(-7 + 3x) + 5(2x + 1)(x + -2) = -1x2 + -1[-1(3x + 1)] (-7 * 2 + 3x * 2) + 5(2x + 1)(x + -2) = -1x2 + -1[-1(3x + 1)] (-14 + 6x) + 5(2x + 1)(x + -2) = -1x2 + -1[-1(3x + 1)] Reorder the terms: -14 + 6x + 5(1 + 2x)(x + -2) = -1x2 + -1[-1(3x + 1)] Reorder the terms: -14 + 6x + 5(1 + 2x)(-2 + x) = -1x2 + -1[-1(3x + 1)] Multiply (1 + 2x) * (-2 + x) -14 + 6x + 5(1(-2 + x) + 2x * (-2 + x)) = -1x2 + -1[-1(3x + 1)] -14 + 6x + 5((-2 * 1 + x * 1) + 2x * (-2 + x)) = -1x2 + -1[-1(3x + 1)] -14 + 6x + 5((-2 + 1x) + 2x * (-2 + x)) = -1x2 + -1[-1(3x + 1)] -14 + 6x + 5(-2 + 1x + (-2 * 2x + x * 2x)) = -1x2 + -1[-1(3x + 1)] -14 + 6x + 5(-2 + 1x + (-4x + 2x2)) = -1x2 + -1[-1(3x + 1)] Combine like terms: 1x + -4x = -3x -14 + 6x + 5(-2 + -3x + 2x2) = -1x2 + -1[-1(3x + 1)] -14 + 6x + (-2 * 5 + -3x * 5 + 2x2 * 5) = -1x2 + -1[-1(3x + 1)] -14 + 6x + (-10 + -15x + 10x2) = -1x2 + -1[-1(3x + 1)] Reorder the terms: -14 + -10 + 6x + -15x + 10x2 = -1x2 + -1[-1(3x + 1)] Combine like terms: -14 + -10 = -24 -24 + 6x + -15x + 10x2 = -1x2 + -1[-1(3x + 1)] Combine like terms: 6x + -15x = -9x -24 + -9x + 10x2 = -1x2 + -1[-1(3x + 1)] Reorder the terms: -24 + -9x + 10x2 = -1x2 + -1[-1(1 + 3x)] -24 + -9x + 10x2 = -1x2 + -1[(1 * -1 + 3x * -1)] -24 + -9x + 10x2 = -1x2 + -1[(-1 + -3x)] -24 + -9x + 10x2 = -1x2 + [-1 * -1 + -3x * -1] -24 + -9x + 10x2 = -1x2 + [1 + 3x] Reorder the terms: -24 + -9x + 10x2 = 1 + 3x + -1x2 Solving -24 + -9x + 10x2 = 1 + 3x + -1x2 Solving for variable 'x'. Reorder the terms: -24 + -1 + -9x + -3x + 10x2 + x2 = 1 + 3x + -1x2 + -1 + -3x + x2 Combine like terms: -24 + -1 = -25 -25 + -9x + -3x + 10x2 + x2 = 1 + 3x + -1x2 + -1 + -3x + x2 Combine like terms: -9x + -3x = -12x -25 + -12x + 10x2 + x2 = 1 + 3x + -1x2 + -1 + -3x + x2 Combine like terms: 10x2 + x2 = 11x2 -25 + -12x + 11x2 = 1 + 3x + -1x2 + -1 + -3x + x2 Reorder the terms: -25 + -12x + 11x2 = 1 + -1 + 3x + -3x + -1x2 + x2 Combine like terms: 1 + -1 = 0 -25 + -12x + 11x2 = 0 + 3x + -3x + -1x2 + x2 -25 + -12x + 11x2 = 3x + -3x + -1x2 + x2 Combine like terms: 3x + -3x = 0 -25 + -12x + 11x2 = 0 + -1x2 + x2 -25 + -12x + 11x2 = -1x2 + x2 Combine like terms: -1x2 + x2 = 0 -25 + -12x + 11x2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -2.272727273 + -1.090909091x + x2 = 0 Move the constant term to the right: Add '2.272727273' to each side of the equation. -2.272727273 + -1.090909091x + 2.272727273 + x2 = 0 + 2.272727273 Reorder the terms: -2.272727273 + 2.272727273 + -1.090909091x + x2 = 0 + 2.272727273 Combine like terms: -2.272727273 + 2.272727273 = 0.000000000 0.000000000 + -1.090909091x + x2 = 0 + 2.272727273 -1.090909091x + x2 = 0 + 2.272727273 Combine like terms: 0 + 2.272727273 = 2.272727273 -1.090909091x + x2 = 2.272727273 The x term is -1.090909091x. Take half its coefficient (-0.5454545455). Square it (0.2975206612) and add it to both sides. Add '0.2975206612' to each side of the equation. -1.090909091x + 0.2975206612 + x2 = 2.272727273 + 0.2975206612 Reorder the terms: 0.2975206612 + -1.090909091x + x2 = 2.272727273 + 0.2975206612 Combine like terms: 2.272727273 + 0.2975206612 = 2.5702479342 0.2975206612 + -1.090909091x + x2 = 2.5702479342 Factor a perfect square on the left side: (x + -0.5454545455)(x + -0.5454545455) = 2.5702479342 Calculate the square root of the right side: 1.603199281 Break this problem into two subproblems by setting (x + -0.5454545455) equal to 1.603199281 and -1.603199281.Subproblem 1
x + -0.5454545455 = 1.603199281 Simplifying x + -0.5454545455 = 1.603199281 Reorder the terms: -0.5454545455 + x = 1.603199281 Solving -0.5454545455 + x = 1.603199281 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5454545455' to each side of the equation. -0.5454545455 + 0.5454545455 + x = 1.603199281 + 0.5454545455 Combine like terms: -0.5454545455 + 0.5454545455 = 0.0000000000 0.0000000000 + x = 1.603199281 + 0.5454545455 x = 1.603199281 + 0.5454545455 Combine like terms: 1.603199281 + 0.5454545455 = 2.1486538265 x = 2.1486538265 Simplifying x = 2.1486538265Subproblem 2
x + -0.5454545455 = -1.603199281 Simplifying x + -0.5454545455 = -1.603199281 Reorder the terms: -0.5454545455 + x = -1.603199281 Solving -0.5454545455 + x = -1.603199281 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5454545455' to each side of the equation. -0.5454545455 + 0.5454545455 + x = -1.603199281 + 0.5454545455 Combine like terms: -0.5454545455 + 0.5454545455 = 0.0000000000 0.0000000000 + x = -1.603199281 + 0.5454545455 x = -1.603199281 + 0.5454545455 Combine like terms: -1.603199281 + 0.5454545455 = -1.0577447355 x = -1.0577447355 Simplifying x = -1.0577447355Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.1486538265, -1.0577447355}
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