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Simplifying (2x2 + -6x + 16x + 8x + -24x + 16)(x2 + 5x + -2) = (2x + 4)(x2 + 5x + -2)(x + 4) Reorder the terms: (16 + -6x + 16x + 8x + -24x + 2x2)(x2 + 5x + -2) = (2x + 4)(x2 + 5x + -2)(x + 4) Combine like terms: -6x + 16x = 10x (16 + 10x + 8x + -24x + 2x2)(x2 + 5x + -2) = (2x + 4)(x2 + 5x + -2)(x + 4) Combine like terms: 10x + 8x = 18x (16 + 18x + -24x + 2x2)(x2 + 5x + -2) = (2x + 4)(x2 + 5x + -2)(x + 4) Combine like terms: 18x + -24x = -6x (16 + -6x + 2x2)(x2 + 5x + -2) = (2x + 4)(x2 + 5x + -2)(x + 4) Reorder the terms: (16 + -6x + 2x2)(-2 + 5x + x2) = (2x + 4)(x2 + 5x + -2)(x + 4) Multiply (16 + -6x + 2x2) * (-2 + 5x + x2) (16(-2 + 5x + x2) + -6x * (-2 + 5x + x2) + 2x2 * (-2 + 5x + x2)) = (2x + 4)(x2 + 5x + -2)(x + 4) ((-2 * 16 + 5x * 16 + x2 * 16) + -6x * (-2 + 5x + x2) + 2x2 * (-2 + 5x + x2)) = (2x + 4)(x2 + 5x + -2)(x + 4) ((-32 + 80x + 16x2) + -6x * (-2 + 5x + x2) + 2x2 * (-2 + 5x + x2)) = (2x + 4)(x2 + 5x + -2)(x + 4) (-32 + 80x + 16x2 + (-2 * -6x + 5x * -6x + x2 * -6x) + 2x2 * (-2 + 5x + x2)) = (2x + 4)(x2 + 5x + -2)(x + 4) (-32 + 80x + 16x2 + (12x + -30x2 + -6x3) + 2x2 * (-2 + 5x + x2)) = (2x + 4)(x2 + 5x + -2)(x + 4) (-32 + 80x + 16x2 + 12x + -30x2 + -6x3 + (-2 * 2x2 + 5x * 2x2 + x2 * 2x2)) = (2x + 4)(x2 + 5x + -2)(x + 4) (-32 + 80x + 16x2 + 12x + -30x2 + -6x3 + (-4x2 + 10x3 + 2x4)) = (2x + 4)(x2 + 5x + -2)(x + 4) Reorder the terms: (-32 + 80x + 12x + 16x2 + -30x2 + -4x2 + -6x3 + 10x3 + 2x4) = (2x + 4)(x2 + 5x + -2)(x + 4) Combine like terms: 80x + 12x = 92x (-32 + 92x + 16x2 + -30x2 + -4x2 + -6x3 + 10x3 + 2x4) = (2x + 4)(x2 + 5x + -2)(x + 4) Combine like terms: 16x2 + -30x2 = -14x2 (-32 + 92x + -14x2 + -4x2 + -6x3 + 10x3 + 2x4) = (2x + 4)(x2 + 5x + -2)(x + 4) Combine like terms: -14x2 + -4x2 = -18x2 (-32 + 92x + -18x2 + -6x3 + 10x3 + 2x4) = (2x + 4)(x2 + 5x + -2)(x + 4) Combine like terms: -6x3 + 10x3 = 4x3 (-32 + 92x + -18x2 + 4x3 + 2x4) = (2x + 4)(x2 + 5x + -2)(x + 4) Reorder the terms: -32 + 92x + -18x2 + 4x3 + 2x4 = (4 + 2x)(x2 + 5x + -2)(x + 4) Reorder the terms: -32 + 92x + -18x2 + 4x3 + 2x4 = (4 + 2x)(-2 + 5x + x2)(x + 4) Reorder the terms: -32 + 92x + -18x2 + 4x3 + 2x4 = (4 + 2x)(-2 + 5x + x2)(4 + x) Multiply (4 + 2x) * (-2 + 5x + x2) -32 + 92x + -18x2 + 4x3 + 2x4 = (4(-2 + 5x + x2) + 2x * (-2 + 5x + x2))(4 + x) -32 + 92x + -18x2 + 4x3 + 2x4 = ((-2 * 4 + 5x * 4 + x2 * 4) + 2x * (-2 + 5x + x2))(4 + x) -32 + 92x + -18x2 + 4x3 + 2x4 = ((-8 + 20x + 4x2) + 2x * (-2 + 5x + x2))(4 + x) -32 + 92x + -18x2 + 4x3 + 2x4 = (-8 + 20x + 4x2 + (-2 * 2x + 5x * 2x + x2 * 2x))(4 + x) -32 + 92x + -18x2 + 4x3 + 2x4 = (-8 + 20x + 4x2 + (-4x + 10x2 + 2x3))(4 + x) Reorder the terms: -32 + 92x + -18x2 + 4x3 + 2x4 = (-8 + 20x + -4x + 4x2 + 10x2 + 2x3)(4 + x) Combine like terms: 20x + -4x = 16x -32 + 92x + -18x2 + 4x3 + 2x4 = (-8 + 16x + 4x2 + 10x2 + 2x3)(4 + x) Combine like terms: 4x2 + 10x2 = 14x2 -32 + 92x + -18x2 + 4x3 + 2x4 = (-8 + 16x + 14x2 + 2x3)(4 + x) Multiply (-8 + 16x + 14x2 + 2x3) * (4 + x) -32 + 92x + -18x2 + 4x3 + 2x4 = (-8(4 + x) + 16x * (4 + x) + 14x2 * (4 + x) + 2x3 * (4 + x)) -32 + 92x + -18x2 + 4x3 + 2x4 = ((4 * -8 + x * -8) + 16x * (4 + x) + 14x2 * (4 + x) + 2x3 * (4 + x)) -32 + 92x + -18x2 + 4x3 + 2x4 = ((-32 + -8x) + 16x * (4 + x) + 14x2 * (4 + x) + 2x3 * (4 + x)) -32 + 92x + -18x2 + 4x3 + 2x4 = (-32 + -8x + (4 * 16x + x * 16x) + 14x2 * (4 + x) + 2x3 * (4 + x)) -32 + 92x + -18x2 + 4x3 + 2x4 = (-32 + -8x + (64x + 16x2) + 14x2 * (4 + x) + 2x3 * (4 + x)) -32 + 92x + -18x2 + 4x3 + 2x4 = (-32 + -8x + 64x + 16x2 + (4 * 14x2 + x * 14x2) + 2x3 * (4 + x)) -32 + 92x + -18x2 + 4x3 + 2x4 = (-32 + -8x + 64x + 16x2 + (56x2 + 14x3) + 2x3 * (4 + x)) -32 + 92x + -18x2 + 4x3 + 2x4 = (-32 + -8x + 64x + 16x2 + 56x2 + 14x3 + (4 * 2x3 + x * 2x3)) -32 + 92x + -18x2 + 4x3 + 2x4 = (-32 + -8x + 64x + 16x2 + 56x2 + 14x3 + (8x3 + 2x4)) Combine like terms: -8x + 64x = 56x -32 + 92x + -18x2 + 4x3 + 2x4 = (-32 + 56x + 16x2 + 56x2 + 14x3 + 8x3 + 2x4) Combine like terms: 16x2 + 56x2 = 72x2 -32 + 92x + -18x2 + 4x3 + 2x4 = (-32 + 56x + 72x2 + 14x3 + 8x3 + 2x4) Combine like terms: 14x3 + 8x3 = 22x3 -32 + 92x + -18x2 + 4x3 + 2x4 = (-32 + 56x + 72x2 + 22x3 + 2x4) Add '32' to each side of the equation. -32 + 92x + -18x2 + 4x3 + 32 + 2x4 = -32 + 56x + 72x2 + 22x3 + 32 + 2x4 Reorder the terms: -32 + 32 + 92x + -18x2 + 4x3 + 2x4 = -32 + 56x + 72x2 + 22x3 + 32 + 2x4 Combine like terms: -32 + 32 = 0 0 + 92x + -18x2 + 4x3 + 2x4 = -32 + 56x + 72x2 + 22x3 + 32 + 2x4 92x + -18x2 + 4x3 + 2x4 = -32 + 56x + 72x2 + 22x3 + 32 + 2x4 Reorder the terms: 92x + -18x2 + 4x3 + 2x4 = -32 + 32 + 56x + 72x2 + 22x3 + 2x4 Combine like terms: -32 + 32 = 0 92x + -18x2 + 4x3 + 2x4 = 0 + 56x + 72x2 + 22x3 + 2x4 92x + -18x2 + 4x3 + 2x4 = 56x + 72x2 + 22x3 + 2x4 Add '-2x4' to each side of the equation. 92x + -18x2 + 4x3 + 2x4 + -2x4 = 56x + 72x2 + 22x3 + 2x4 + -2x4 Combine like terms: 2x4 + -2x4 = 0 92x + -18x2 + 4x3 + 0 = 56x + 72x2 + 22x3 + 2x4 + -2x4 92x + -18x2 + 4x3 = 56x + 72x2 + 22x3 + 2x4 + -2x4 Combine like terms: 2x4 + -2x4 = 0 92x + -18x2 + 4x3 = 56x + 72x2 + 22x3 + 0 92x + -18x2 + 4x3 = 56x + 72x2 + 22x3 Solving 92x + -18x2 + 4x3 = 56x + 72x2 + 22x3 Solving for variable 'x'. Reorder the terms: 92x + -56x + -18x2 + -72x2 + 4x3 + -22x3 = 56x + 72x2 + 22x3 + -56x + -72x2 + -22x3 Combine like terms: 92x + -56x = 36x 36x + -18x2 + -72x2 + 4x3 + -22x3 = 56x + 72x2 + 22x3 + -56x + -72x2 + -22x3 Combine like terms: -18x2 + -72x2 = -90x2 36x + -90x2 + 4x3 + -22x3 = 56x + 72x2 + 22x3 + -56x + -72x2 + -22x3 Combine like terms: 4x3 + -22x3 = -18x3 36x + -90x2 + -18x3 = 56x + 72x2 + 22x3 + -56x + -72x2 + -22x3 Reorder the terms: 36x + -90x2 + -18x3 = 56x + -56x + 72x2 + -72x2 + 22x3 + -22x3 Combine like terms: 56x + -56x = 0 36x + -90x2 + -18x3 = 0 + 72x2 + -72x2 + 22x3 + -22x3 36x + -90x2 + -18x3 = 72x2 + -72x2 + 22x3 + -22x3 Combine like terms: 72x2 + -72x2 = 0 36x + -90x2 + -18x3 = 0 + 22x3 + -22x3 36x + -90x2 + -18x3 = 22x3 + -22x3 Combine like terms: 22x3 + -22x3 = 0 36x + -90x2 + -18x3 = 0 Factor out the Greatest Common Factor (GCF), '18x'. 18x(2 + -5x + -1x2) = 0 Ignore the factor 18.Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x = 0Subproblem 2
Set the factor '(2 + -5x + -1x2)' equal to zero and attempt to solve: Simplifying 2 + -5x + -1x2 = 0 Solving 2 + -5x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -2 + 5x + x2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 5x + 2 + x2 = 0 + 2 Reorder the terms: -2 + 2 + 5x + x2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 5x + x2 = 0 + 2 5x + x2 = 0 + 2 Combine like terms: 0 + 2 = 2 5x + x2 = 2 The x term is 5x. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5x + 6.25 + x2 = 2 + 6.25 Reorder the terms: 6.25 + 5x + x2 = 2 + 6.25 Combine like terms: 2 + 6.25 = 8.25 6.25 + 5x + x2 = 8.25 Factor a perfect square on the left side: (x + 2.5)(x + 2.5) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (x + 2.5) equal to 2.872281323 and -2.872281323.Subproblem 1
x + 2.5 = 2.872281323 Simplifying x + 2.5 = 2.872281323 Reorder the terms: 2.5 + x = 2.872281323 Solving 2.5 + x = 2.872281323 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 2.872281323 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 2.872281323 + -2.5 x = 2.872281323 + -2.5 Combine like terms: 2.872281323 + -2.5 = 0.372281323 x = 0.372281323 Simplifying x = 0.372281323Subproblem 2
x + 2.5 = -2.872281323 Simplifying x + 2.5 = -2.872281323 Reorder the terms: 2.5 + x = -2.872281323 Solving 2.5 + x = -2.872281323 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -2.872281323 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -2.872281323 + -2.5 x = -2.872281323 + -2.5 Combine like terms: -2.872281323 + -2.5 = -5.372281323 x = -5.372281323 Simplifying x = -5.372281323Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.372281323, -5.372281323}Solution
x = {0, 0.372281323, -5.372281323}
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