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Simplifying (5x3 + -6x2 + 4x) + (7x3 + 9x2 + -10x) = 0 Reorder the terms: (4x + -6x2 + 5x3) + (7x3 + 9x2 + -10x) = 0 Remove parenthesis around (4x + -6x2 + 5x3) 4x + -6x2 + 5x3 + (7x3 + 9x2 + -10x) = 0 Reorder the terms: 4x + -6x2 + 5x3 + (-10x + 9x2 + 7x3) = 0 Remove parenthesis around (-10x + 9x2 + 7x3) 4x + -6x2 + 5x3 + -10x + 9x2 + 7x3 = 0 Reorder the terms: 4x + -10x + -6x2 + 9x2 + 5x3 + 7x3 = 0 Combine like terms: 4x + -10x = -6x -6x + -6x2 + 9x2 + 5x3 + 7x3 = 0 Combine like terms: -6x2 + 9x2 = 3x2 -6x + 3x2 + 5x3 + 7x3 = 0 Combine like terms: 5x3 + 7x3 = 12x3 -6x + 3x2 + 12x3 = 0 Solving -6x + 3x2 + 12x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '3x'. 3x(-2 + x + 4x2) = 0 Ignore the factor 3.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 + x + 4x2)' equal to zero and attempt to solve: Simplifying -2 + x + 4x2 = 0 Solving -2 + x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.5 + 0.25x + x2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 0.25x + 0.5 + x2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 0.25x + x2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 0.25x + x2 = 0 + 0.5 0.25x + x2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 0.25x + x2 = 0.5 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.25x + 0.25 + x2 = 0.5 + 0.25 Reorder the terms: 0.25 + 0.25x + x2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 0.25 + 0.25x + x2 = 0.75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + 0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
x + 0.5 = 0.866025404 Simplifying x + 0.5 = 0.866025404 Reorder the terms: 0.5 + x = 0.866025404 Solving 0.5 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.866025404 + -0.5 x = 0.866025404 + -0.5 Combine like terms: 0.866025404 + -0.5 = 0.366025404 x = 0.366025404 Simplifying x = 0.366025404Subproblem 2
x + 0.5 = -0.866025404 Simplifying x + 0.5 = -0.866025404 Reorder the terms: 0.5 + x = -0.866025404 Solving 0.5 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.866025404 + -0.5 x = -0.866025404 + -0.5 Combine like terms: -0.866025404 + -0.5 = -1.366025404 x = -1.366025404 Simplifying x = -1.366025404Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.366025404, -1.366025404}Solution
x = {0, 0.366025404, -1.366025404}
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