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Simplifying (10 + -2x)(25 + -6x)(15 + -2x) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 Multiply (10 + -2x) * (25 + -6x) (10(25 + -6x) + -2x * (25 + -6x))(15 + -2x) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 ((25 * 10 + -6x * 10) + -2x * (25 + -6x))(15 + -2x) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 ((250 + -60x) + -2x * (25 + -6x))(15 + -2x) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 (250 + -60x + (25 * -2x + -6x * -2x))(15 + -2x) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 (250 + -60x + (-50x + 12x2))(15 + -2x) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 Combine like terms: -60x + -50x = -110x (250 + -110x + 12x2)(15 + -2x) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 Multiply (250 + -110x + 12x2) * (15 + -2x) (250(15 + -2x) + -110x * (15 + -2x) + 12x2 * (15 + -2x)) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 ((15 * 250 + -2x * 250) + -110x * (15 + -2x) + 12x2 * (15 + -2x)) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 ((3750 + -500x) + -110x * (15 + -2x) + 12x2 * (15 + -2x)) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 (3750 + -500x + (15 * -110x + -2x * -110x) + 12x2 * (15 + -2x)) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 (3750 + -500x + (-1650x + 220x2) + 12x2 * (15 + -2x)) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 (3750 + -500x + -1650x + 220x2 + (15 * 12x2 + -2x * 12x2)) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 (3750 + -500x + -1650x + 220x2 + (180x2 + -24x3)) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 Combine like terms: -500x + -1650x = -2150x (3750 + -2150x + 220x2 + 180x2 + -24x3) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 Combine like terms: 220x2 + 180x2 = 400x2 (3750 + -2150x + 400x2 + -24x3) + -1(225(10 + -2x) + 100(15 + -2x)) = 0 3750 + -2150x + 400x2 + -24x3 + -1((10 * 225 + -2x * 225) + 100(15 + -2x)) = 0 3750 + -2150x + 400x2 + -24x3 + -1((2250 + -450x) + 100(15 + -2x)) = 0 3750 + -2150x + 400x2 + -24x3 + -1(2250 + -450x + (15 * 100 + -2x * 100)) = 0 3750 + -2150x + 400x2 + -24x3 + -1(2250 + -450x + (1500 + -200x)) = 0 Reorder the terms: 3750 + -2150x + 400x2 + -24x3 + -1(2250 + 1500 + -450x + -200x) = 0 Combine like terms: 2250 + 1500 = 3750 3750 + -2150x + 400x2 + -24x3 + -1(3750 + -450x + -200x) = 0 Combine like terms: -450x + -200x = -650x 3750 + -2150x + 400x2 + -24x3 + -1(3750 + -650x) = 0 3750 + -2150x + 400x2 + -24x3 + (3750 * -1 + -650x * -1) = 0 3750 + -2150x + 400x2 + -24x3 + (-3750 + 650x) = 0 Reorder the terms: 3750 + -3750 + -2150x + 650x + 400x2 + -24x3 = 0 Combine like terms: 3750 + -3750 = 0 0 + -2150x + 650x + 400x2 + -24x3 = 0 -2150x + 650x + 400x2 + -24x3 = 0 Combine like terms: -2150x + 650x = -1500x -1500x + 400x2 + -24x3 = 0 Solving -1500x + 400x2 + -24x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '4x'. 4x(-375 + 100x + -6x2) = 0 Ignore the factor 4.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 '(-375 + 100x + -6x2)' equal to zero and attempt to solve: Simplifying -375 + 100x + -6x2 = 0 Solving -375 + 100x + -6x2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. 62.5 + -16.66666667x + x2 = 0 Move the constant term to the right: Add '-62.5' to each side of the equation. 62.5 + -16.66666667x + -62.5 + x2 = 0 + -62.5 Reorder the terms: 62.5 + -62.5 + -16.66666667x + x2 = 0 + -62.5 Combine like terms: 62.5 + -62.5 = 0.0 0.0 + -16.66666667x + x2 = 0 + -62.5 -16.66666667x + x2 = 0 + -62.5 Combine like terms: 0 + -62.5 = -62.5 -16.66666667x + x2 = -62.5 The x term is -16.66666667x. Take half its coefficient (-8.333333335). Square it (69.44444447) and add it to both sides. Add '69.44444447' to each side of the equation. -16.66666667x + 69.44444447 + x2 = -62.5 + 69.44444447 Reorder the terms: 69.44444447 + -16.66666667x + x2 = -62.5 + 69.44444447 Combine like terms: -62.5 + 69.44444447 = 6.94444447 69.44444447 + -16.66666667x + x2 = 6.94444447 Factor a perfect square on the left side: (x + -8.333333335)(x + -8.333333335) = 6.94444447 Calculate the square root of the right side: 2.635231388 Break this problem into two subproblems by setting (x + -8.333333335) equal to 2.635231388 and -2.635231388.Subproblem 1
x + -8.333333335 = 2.635231388 Simplifying x + -8.333333335 = 2.635231388 Reorder the terms: -8.333333335 + x = 2.635231388 Solving -8.333333335 + x = 2.635231388 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8.333333335' to each side of the equation. -8.333333335 + 8.333333335 + x = 2.635231388 + 8.333333335 Combine like terms: -8.333333335 + 8.333333335 = 0.000000000 0.000000000 + x = 2.635231388 + 8.333333335 x = 2.635231388 + 8.333333335 Combine like terms: 2.635231388 + 8.333333335 = 10.968564723 x = 10.968564723 Simplifying x = 10.968564723Subproblem 2
x + -8.333333335 = -2.635231388 Simplifying x + -8.333333335 = -2.635231388 Reorder the terms: -8.333333335 + x = -2.635231388 Solving -8.333333335 + x = -2.635231388 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8.333333335' to each side of the equation. -8.333333335 + 8.333333335 + x = -2.635231388 + 8.333333335 Combine like terms: -8.333333335 + 8.333333335 = 0.000000000 0.000000000 + x = -2.635231388 + 8.333333335 x = -2.635231388 + 8.333333335 Combine like terms: -2.635231388 + 8.333333335 = 5.698101947 x = 5.698101947 Simplifying x = 5.698101947Solution
The solution to the problem is based on the solutions from the subproblems. x = {10.968564723, 5.698101947}Solution
x = {0, 10.968564723, 5.698101947}
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