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Simplifying 12u3 + 42u2 + -40u = 0 Reorder the terms: -40u + 42u2 + 12u3 = 0 Solving -40u + 42u2 + 12u3 = 0 Solving for variable 'u'. Factor out the Greatest Common Factor (GCF), '2u'. 2u(-20 + 21u + 6u2) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'u' equal to zero and attempt to solve: Simplifying u = 0 Solving u = 0 Move all terms containing u to the left, all other terms to the right. Simplifying u = 0Subproblem 2
Set the factor '(-20 + 21u + 6u2)' equal to zero and attempt to solve: Simplifying -20 + 21u + 6u2 = 0 Solving -20 + 21u + 6u2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -3.333333333 + 3.5u + u2 = 0 Move the constant term to the right: Add '3.333333333' to each side of the equation. -3.333333333 + 3.5u + 3.333333333 + u2 = 0 + 3.333333333 Reorder the terms: -3.333333333 + 3.333333333 + 3.5u + u2 = 0 + 3.333333333 Combine like terms: -3.333333333 + 3.333333333 = 0.000000000 0.000000000 + 3.5u + u2 = 0 + 3.333333333 3.5u + u2 = 0 + 3.333333333 Combine like terms: 0 + 3.333333333 = 3.333333333 3.5u + u2 = 3.333333333 The u term is 3.5u. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5u + 3.0625 + u2 = 3.333333333 + 3.0625 Reorder the terms: 3.0625 + 3.5u + u2 = 3.333333333 + 3.0625 Combine like terms: 3.333333333 + 3.0625 = 6.395833333 3.0625 + 3.5u + u2 = 6.395833333 Factor a perfect square on the left side: (u + 1.75)(u + 1.75) = 6.395833333 Calculate the square root of the right side: 2.528998484 Break this problem into two subproblems by setting (u + 1.75) equal to 2.528998484 and -2.528998484.Subproblem 1
u + 1.75 = 2.528998484 Simplifying u + 1.75 = 2.528998484 Reorder the terms: 1.75 + u = 2.528998484 Solving 1.75 + u = 2.528998484 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + u = 2.528998484 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + u = 2.528998484 + -1.75 u = 2.528998484 + -1.75 Combine like terms: 2.528998484 + -1.75 = 0.778998484 u = 0.778998484 Simplifying u = 0.778998484Subproblem 2
u + 1.75 = -2.528998484 Simplifying u + 1.75 = -2.528998484 Reorder the terms: 1.75 + u = -2.528998484 Solving 1.75 + u = -2.528998484 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + u = -2.528998484 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + u = -2.528998484 + -1.75 u = -2.528998484 + -1.75 Combine like terms: -2.528998484 + -1.75 = -4.278998484 u = -4.278998484 Simplifying u = -4.278998484Solution
The solution to the problem is based on the solutions from the subproblems. u = {0.778998484, -4.278998484}Solution
u = {0, 0.778998484, -4.278998484}
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