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Simplifying x(x + 1)(x + -5) + (x + -2) * 2 = x(x + 3)(x + -3) + -1x(x + 4) + -2 Reorder the terms: x(1 + x)(x + -5) + (x + -2) * 2 = x(x + 3)(x + -3) + -1x(x + 4) + -2 Reorder the terms: x(1 + x)(-5 + x) + (x + -2) * 2 = x(x + 3)(x + -3) + -1x(x + 4) + -2 Multiply (1 + x) * (-5 + x) x(1(-5 + x) + x(-5 + x)) + (x + -2) * 2 = x(x + 3)(x + -3) + -1x(x + 4) + -2 x((-5 * 1 + x * 1) + x(-5 + x)) + (x + -2) * 2 = x(x + 3)(x + -3) + -1x(x + 4) + -2 x((-5 + 1x) + x(-5 + x)) + (x + -2) * 2 = x(x + 3)(x + -3) + -1x(x + 4) + -2 x(-5 + 1x + (-5 * x + x * x)) + (x + -2) * 2 = x(x + 3)(x + -3) + -1x(x + 4) + -2 x(-5 + 1x + (-5x + x2)) + (x + -2) * 2 = x(x + 3)(x + -3) + -1x(x + 4) + -2 Combine like terms: 1x + -5x = -4x x(-5 + -4x + x2) + (x + -2) * 2 = x(x + 3)(x + -3) + -1x(x + 4) + -2 (-5 * x + -4x * x + x2 * x) + (x + -2) * 2 = x(x + 3)(x + -3) + -1x(x + 4) + -2 (-5x + -4x2 + x3) + (x + -2) * 2 = x(x + 3)(x + -3) + -1x(x + 4) + -2 Reorder the terms: -5x + -4x2 + x3 + (-2 + x) * 2 = x(x + 3)(x + -3) + -1x(x + 4) + -2 Reorder the terms for easier multiplication: -5x + -4x2 + x3 + 2(-2 + x) = x(x + 3)(x + -3) + -1x(x + 4) + -2 -5x + -4x2 + x3 + (-2 * 2 + x * 2) = x(x + 3)(x + -3) + -1x(x + 4) + -2 -5x + -4x2 + x3 + (-4 + 2x) = x(x + 3)(x + -3) + -1x(x + 4) + -2 Reorder the terms: -4 + -5x + 2x + -4x2 + x3 = x(x + 3)(x + -3) + -1x(x + 4) + -2 Combine like terms: -5x + 2x = -3x -4 + -3x + -4x2 + x3 = x(x + 3)(x + -3) + -1x(x + 4) + -2 Reorder the terms: -4 + -3x + -4x2 + x3 = x(3 + x)(x + -3) + -1x(x + 4) + -2 Reorder the terms: -4 + -3x + -4x2 + x3 = x(3 + x)(-3 + x) + -1x(x + 4) + -2 Multiply (3 + x) * (-3 + x) -4 + -3x + -4x2 + x3 = x(3(-3 + x) + x(-3 + x)) + -1x(x + 4) + -2 -4 + -3x + -4x2 + x3 = x((-3 * 3 + x * 3) + x(-3 + x)) + -1x(x + 4) + -2 -4 + -3x + -4x2 + x3 = x((-9 + 3x) + x(-3 + x)) + -1x(x + 4) + -2 -4 + -3x + -4x2 + x3 = x(-9 + 3x + (-3 * x + x * x)) + -1x(x + 4) + -2 -4 + -3x + -4x2 + x3 = x(-9 + 3x + (-3x + x2)) + -1x(x + 4) + -2 Combine like terms: 3x + -3x = 0 -4 + -3x + -4x2 + x3 = x(-9 + 0 + x2) + -1x(x + 4) + -2 -4 + -3x + -4x2 + x3 = x(-9 + x2) + -1x(x + 4) + -2 -4 + -3x + -4x2 + x3 = (-9 * x + x2 * x) + -1x(x + 4) + -2 -4 + -3x + -4x2 + x3 = (-9x + x3) + -1x(x + 4) + -2 Reorder the terms: -4 + -3x + -4x2 + x3 = -9x + x3 + -1x(4 + x) + -2 -4 + -3x + -4x2 + x3 = -9x + x3 + (4 * -1x + x * -1x) + -2 -4 + -3x + -4x2 + x3 = -9x + x3 + (-4x + -1x2) + -2 Reorder the terms: -4 + -3x + -4x2 + x3 = -2 + -9x + -4x + -1x2 + x3 Combine like terms: -9x + -4x = -13x -4 + -3x + -4x2 + x3 = -2 + -13x + -1x2 + x3 Add '-1x3' to each side of the equation. -4 + -3x + -4x2 + x3 + -1x3 = -2 + -13x + -1x2 + x3 + -1x3 Combine like terms: x3 + -1x3 = 0 -4 + -3x + -4x2 + 0 = -2 + -13x + -1x2 + x3 + -1x3 -4 + -3x + -4x2 = -2 + -13x + -1x2 + x3 + -1x3 Combine like terms: x3 + -1x3 = 0 -4 + -3x + -4x2 = -2 + -13x + -1x2 + 0 -4 + -3x + -4x2 = -2 + -13x + -1x2 Solving -4 + -3x + -4x2 = -2 + -13x + -1x2 Solving for variable 'x'. Reorder the terms: -4 + 2 + -3x + 13x + -4x2 + x2 = -2 + -13x + -1x2 + 2 + 13x + x2 Combine like terms: -4 + 2 = -2 -2 + -3x + 13x + -4x2 + x2 = -2 + -13x + -1x2 + 2 + 13x + x2 Combine like terms: -3x + 13x = 10x -2 + 10x + -4x2 + x2 = -2 + -13x + -1x2 + 2 + 13x + x2 Combine like terms: -4x2 + x2 = -3x2 -2 + 10x + -3x2 = -2 + -13x + -1x2 + 2 + 13x + x2 Reorder the terms: -2 + 10x + -3x2 = -2 + 2 + -13x + 13x + -1x2 + x2 Combine like terms: -2 + 2 = 0 -2 + 10x + -3x2 = 0 + -13x + 13x + -1x2 + x2 -2 + 10x + -3x2 = -13x + 13x + -1x2 + x2 Combine like terms: -13x + 13x = 0 -2 + 10x + -3x2 = 0 + -1x2 + x2 -2 + 10x + -3x2 = -1x2 + x2 Combine like terms: -1x2 + x2 = 0 -2 + 10x + -3x2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 0.6666666667 + -3.333333333x + x2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + -3.333333333x + -0.6666666667 + x2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + -3.333333333x + x2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + -3.333333333x + x2 = 0 + -0.6666666667 -3.333333333x + x2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 -3.333333333x + x2 = -0.6666666667 The x term is -3.333333333x. Take half its coefficient (-1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. -3.333333333x + 2.777777779 + x2 = -0.6666666667 + 2.777777779 Reorder the terms: 2.777777779 + -3.333333333x + x2 = -0.6666666667 + 2.777777779 Combine like terms: -0.6666666667 + 2.777777779 = 2.1111111123 2.777777779 + -3.333333333x + x2 = 2.1111111123 Factor a perfect square on the left side: (x + -1.666666667)(x + -1.666666667) = 2.1111111123 Calculate the square root of the right side: 1.452966315 Break this problem into two subproblems by setting (x + -1.666666667) equal to 1.452966315 and -1.452966315.Subproblem 1
x + -1.666666667 = 1.452966315 Simplifying x + -1.666666667 = 1.452966315 Reorder the terms: -1.666666667 + x = 1.452966315 Solving -1.666666667 + x = 1.452966315 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + x = 1.452966315 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + x = 1.452966315 + 1.666666667 x = 1.452966315 + 1.666666667 Combine like terms: 1.452966315 + 1.666666667 = 3.119632982 x = 3.119632982 Simplifying x = 3.119632982Subproblem 2
x + -1.666666667 = -1.452966315 Simplifying x + -1.666666667 = -1.452966315 Reorder the terms: -1.666666667 + x = -1.452966315 Solving -1.666666667 + x = -1.452966315 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + x = -1.452966315 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + x = -1.452966315 + 1.666666667 x = -1.452966315 + 1.666666667 Combine like terms: -1.452966315 + 1.666666667 = 0.213700352 x = 0.213700352 Simplifying x = 0.213700352Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.119632982, 0.213700352}
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