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Simplifying (2u + 3)(u + -4) + 4u(u + -2) = 0 Reorder the terms: (3 + 2u)(u + -4) + 4u(u + -2) = 0 Reorder the terms: (3 + 2u)(-4 + u) + 4u(u + -2) = 0 Multiply (3 + 2u) * (-4 + u) (3(-4 + u) + 2u * (-4 + u)) + 4u(u + -2) = 0 ((-4 * 3 + u * 3) + 2u * (-4 + u)) + 4u(u + -2) = 0 ((-12 + 3u) + 2u * (-4 + u)) + 4u(u + -2) = 0 (-12 + 3u + (-4 * 2u + u * 2u)) + 4u(u + -2) = 0 (-12 + 3u + (-8u + 2u2)) + 4u(u + -2) = 0 Combine like terms: 3u + -8u = -5u (-12 + -5u + 2u2) + 4u(u + -2) = 0 Reorder the terms: -12 + -5u + 2u2 + 4u(-2 + u) = 0 -12 + -5u + 2u2 + (-2 * 4u + u * 4u) = 0 -12 + -5u + 2u2 + (-8u + 4u2) = 0 Reorder the terms: -12 + -5u + -8u + 2u2 + 4u2 = 0 Combine like terms: -5u + -8u = -13u -12 + -13u + 2u2 + 4u2 = 0 Combine like terms: 2u2 + 4u2 = 6u2 -12 + -13u + 6u2 = 0 Solving -12 + -13u + 6u2 = 0 Solving for variable 'u'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -2 + -2.166666667u + u2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -2.166666667u + 2 + u2 = 0 + 2 Reorder the terms: -2 + 2 + -2.166666667u + u2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -2.166666667u + u2 = 0 + 2 -2.166666667u + u2 = 0 + 2 Combine like terms: 0 + 2 = 2 -2.166666667u + u2 = 2 The u term is -2.166666667u. Take half its coefficient (-1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. -2.166666667u + 1.173611113 + u2 = 2 + 1.173611113 Reorder the terms: 1.173611113 + -2.166666667u + u2 = 2 + 1.173611113 Combine like terms: 2 + 1.173611113 = 3.173611113 1.173611113 + -2.166666667u + u2 = 3.173611113 Factor a perfect square on the left side: (u + -1.083333334)(u + -1.083333334) = 3.173611113 Calculate the square root of the right side: 1.781463194 Break this problem into two subproblems by setting (u + -1.083333334) equal to 1.781463194 and -1.781463194.Subproblem 1
u + -1.083333334 = 1.781463194 Simplifying u + -1.083333334 = 1.781463194 Reorder the terms: -1.083333334 + u = 1.781463194 Solving -1.083333334 + u = 1.781463194 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + u = 1.781463194 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + u = 1.781463194 + 1.083333334 u = 1.781463194 + 1.083333334 Combine like terms: 1.781463194 + 1.083333334 = 2.864796528 u = 2.864796528 Simplifying u = 2.864796528Subproblem 2
u + -1.083333334 = -1.781463194 Simplifying u + -1.083333334 = -1.781463194 Reorder the terms: -1.083333334 + u = -1.781463194 Solving -1.083333334 + u = -1.781463194 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + u = -1.781463194 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + u = -1.781463194 + 1.083333334 u = -1.781463194 + 1.083333334 Combine like terms: -1.781463194 + 1.083333334 = -0.69812986 u = -0.69812986 Simplifying u = -0.69812986Solution
The solution to the problem is based on the solutions from the subproblems. u = {2.864796528, -0.69812986}
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