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Simplifying 10x4 + -30x3 + 15x2 = 0 Reorder the terms: 15x2 + -30x3 + 10x4 = 0 Solving 15x2 + -30x3 + 10x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '5x2'. 5x2(3 + -6x + 2x2) = 0 Ignore the factor 5.Subproblem 1
Set the factor 'x2' equal to zero and attempt to solve: Simplifying x2 = 0 Solving x2 = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x2 = 0 Take the square root of each side: x = {0}Subproblem 2
Set the factor '(3 + -6x + 2x2)' equal to zero and attempt to solve: Simplifying 3 + -6x + 2x2 = 0 Solving 3 + -6x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + -3x + x2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -3x + -1.5 + x2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -3x + x2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -3x + x2 = 0 + -1.5 -3x + x2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -3x + x2 = -1.5 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 = -1.5 + 2.25 Reorder the terms: 2.25 + -3x + x2 = -1.5 + 2.25 Combine like terms: -1.5 + 2.25 = 0.75 2.25 + -3x + x2 = 0.75 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + -1.5) equal to 0.866025404 and -0.866025404.Subproblem 1
x + -1.5 = 0.866025404 Simplifying x + -1.5 = 0.866025404 Reorder the terms: -1.5 + x = 0.866025404 Solving -1.5 + x = 0.866025404 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 = 0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 0.866025404 + 1.5 x = 0.866025404 + 1.5 Combine like terms: 0.866025404 + 1.5 = 2.366025404 x = 2.366025404 Simplifying x = 2.366025404Subproblem 2
x + -1.5 = -0.866025404 Simplifying x + -1.5 = -0.866025404 Reorder the terms: -1.5 + x = -0.866025404 Solving -1.5 + x = -0.866025404 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 = -0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -0.866025404 + 1.5 x = -0.866025404 + 1.5 Combine like terms: -0.866025404 + 1.5 = 0.633974596 x = 0.633974596 Simplifying x = 0.633974596Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.366025404, 0.633974596}Solution
x = {0, 2.366025404, 0.633974596}
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