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Simplifying x2 + -6x + 8(x + 3) = 3x3 + -24x2 + 8x + 24 Reorder the terms: x2 + -6x + 8(3 + x) = 3x3 + -24x2 + 8x + 24 x2 + -6x + (3 * 8 + x * 8) = 3x3 + -24x2 + 8x + 24 x2 + -6x + (24 + 8x) = 3x3 + -24x2 + 8x + 24 Reorder the terms: 24 + -6x + 8x + x2 = 3x3 + -24x2 + 8x + 24 Combine like terms: -6x + 8x = 2x 24 + 2x + x2 = 3x3 + -24x2 + 8x + 24 Reorder the terms: 24 + 2x + x2 = 24 + 8x + -24x2 + 3x3 Add '-24' to each side of the equation. 24 + 2x + -24 + x2 = 24 + 8x + -24x2 + -24 + 3x3 Reorder the terms: 24 + -24 + 2x + x2 = 24 + 8x + -24x2 + -24 + 3x3 Combine like terms: 24 + -24 = 0 0 + 2x + x2 = 24 + 8x + -24x2 + -24 + 3x3 2x + x2 = 24 + 8x + -24x2 + -24 + 3x3 Reorder the terms: 2x + x2 = 24 + -24 + 8x + -24x2 + 3x3 Combine like terms: 24 + -24 = 0 2x + x2 = 0 + 8x + -24x2 + 3x3 2x + x2 = 8x + -24x2 + 3x3 Solving 2x + x2 = 8x + -24x2 + 3x3 Solving for variable 'x'. Reorder the terms: 2x + -8x + x2 + 24x2 + -3x3 = 8x + -24x2 + 3x3 + -8x + 24x2 + -3x3 Combine like terms: 2x + -8x = -6x -6x + x2 + 24x2 + -3x3 = 8x + -24x2 + 3x3 + -8x + 24x2 + -3x3 Combine like terms: x2 + 24x2 = 25x2 -6x + 25x2 + -3x3 = 8x + -24x2 + 3x3 + -8x + 24x2 + -3x3 Reorder the terms: -6x + 25x2 + -3x3 = 8x + -8x + -24x2 + 24x2 + 3x3 + -3x3 Combine like terms: 8x + -8x = 0 -6x + 25x2 + -3x3 = 0 + -24x2 + 24x2 + 3x3 + -3x3 -6x + 25x2 + -3x3 = -24x2 + 24x2 + 3x3 + -3x3 Combine like terms: -24x2 + 24x2 = 0 -6x + 25x2 + -3x3 = 0 + 3x3 + -3x3 -6x + 25x2 + -3x3 = 3x3 + -3x3 Combine like terms: 3x3 + -3x3 = 0 -6x + 25x2 + -3x3 = 0 Factor out the Greatest Common Factor (GCF), 'x'. x(-6 + 25x + -3x2) = 0Subproblem 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 '(-6 + 25x + -3x2)' equal to zero and attempt to solve: Simplifying -6 + 25x + -3x2 = 0 Solving -6 + 25x + -3x2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 2 + -8.333333333x + x2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + -8.333333333x + -2 + x2 = 0 + -2 Reorder the terms: 2 + -2 + -8.333333333x + x2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -8.333333333x + x2 = 0 + -2 -8.333333333x + x2 = 0 + -2 Combine like terms: 0 + -2 = -2 -8.333333333x + x2 = -2 The x term is -8.333333333x. Take half its coefficient (-4.166666667). Square it (17.36111111) and add it to both sides. Add '17.36111111' to each side of the equation. -8.333333333x + 17.36111111 + x2 = -2 + 17.36111111 Reorder the terms: 17.36111111 + -8.333333333x + x2 = -2 + 17.36111111 Combine like terms: -2 + 17.36111111 = 15.36111111 17.36111111 + -8.333333333x + x2 = 15.36111111 Factor a perfect square on the left side: (x + -4.166666667)(x + -4.166666667) = 15.36111111 Calculate the square root of the right side: 3.919325339 Break this problem into two subproblems by setting (x + -4.166666667) equal to 3.919325339 and -3.919325339.Subproblem 1
x + -4.166666667 = 3.919325339 Simplifying x + -4.166666667 = 3.919325339 Reorder the terms: -4.166666667 + x = 3.919325339 Solving -4.166666667 + x = 3.919325339 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.166666667' to each side of the equation. -4.166666667 + 4.166666667 + x = 3.919325339 + 4.166666667 Combine like terms: -4.166666667 + 4.166666667 = 0.000000000 0.000000000 + x = 3.919325339 + 4.166666667 x = 3.919325339 + 4.166666667 Combine like terms: 3.919325339 + 4.166666667 = 8.085992006 x = 8.085992006 Simplifying x = 8.085992006Subproblem 2
x + -4.166666667 = -3.919325339 Simplifying x + -4.166666667 = -3.919325339 Reorder the terms: -4.166666667 + x = -3.919325339 Solving -4.166666667 + x = -3.919325339 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.166666667' to each side of the equation. -4.166666667 + 4.166666667 + x = -3.919325339 + 4.166666667 Combine like terms: -4.166666667 + 4.166666667 = 0.000000000 0.000000000 + x = -3.919325339 + 4.166666667 x = -3.919325339 + 4.166666667 Combine like terms: -3.919325339 + 4.166666667 = 0.247341328 x = 0.247341328 Simplifying x = 0.247341328Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.085992006, 0.247341328}Solution
x = {0, 8.085992006, 0.247341328}
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