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Simplifying 5x3 + 3x2 + -6x3 + 24x = 24x + x Reorder the terms: 24x + 3x2 + 5x3 + -6x3 = 24x + x Combine like terms: 5x3 + -6x3 = -1x3 24x + 3x2 + -1x3 = 24x + x Combine like terms: 24x + x = 25x 24x + 3x2 + -1x3 = 25x Solving 24x + 3x2 + -1x3 = 25x Solving for variable 'x'. Reorder the terms: 24x + -25x + 3x2 + -1x3 = 25x + -25x Combine like terms: 24x + -25x = -1x -1x + 3x2 + -1x3 = 25x + -25x Combine like terms: 25x + -25x = 0 -1x + 3x2 + -1x3 = 0 Factor out the Greatest Common Factor (GCF), 'x'. x(-1 + 3x + -1x2) = 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 '(-1 + 3x + -1x2)' equal to zero and attempt to solve: Simplifying -1 + 3x + -1x2 = 0 Solving -1 + 3x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 1 + -3x + x2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -3x + -1 + x2 = 0 + -1 Reorder the terms: 1 + -1 + -3x + x2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -3x + x2 = 0 + -1 -3x + x2 = 0 + -1 Combine like terms: 0 + -1 = -1 -3x + x2 = -1 The x term is -3x. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3x + 2.25 + x2 = -1 + 2.25 Reorder the terms: 2.25 + -3x + x2 = -1 + 2.25 Combine like terms: -1 + 2.25 = 1.25 2.25 + -3x + x2 = 1.25 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (x + -1.5) equal to 1.118033989 and -1.118033989.Subproblem 1
x + -1.5 = 1.118033989 Simplifying x + -1.5 = 1.118033989 Reorder the terms: -1.5 + x = 1.118033989 Solving -1.5 + x = 1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 1.118033989 + 1.5 x = 1.118033989 + 1.5 Combine like terms: 1.118033989 + 1.5 = 2.618033989 x = 2.618033989 Simplifying x = 2.618033989Subproblem 2
x + -1.5 = -1.118033989 Simplifying x + -1.5 = -1.118033989 Reorder the terms: -1.5 + x = -1.118033989 Solving -1.5 + x = -1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -1.118033989 + 1.5 x = -1.118033989 + 1.5 Combine like terms: -1.118033989 + 1.5 = 0.381966011 x = 0.381966011 Simplifying x = 0.381966011Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.618033989, 0.381966011}Solution
x = {0, 2.618033989, 0.381966011}
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