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Simplifying 2u2 + -13u + 20 = (u + 4)(u + 4) Reorder the terms: 20 + -13u + 2u2 = (u + 4)(u + 4) Reorder the terms: 20 + -13u + 2u2 = (4 + u)(u + 4) Reorder the terms: 20 + -13u + 2u2 = (4 + u)(4 + u) Multiply (4 + u) * (4 + u) 20 + -13u + 2u2 = (4(4 + u) + u(4 + u)) 20 + -13u + 2u2 = ((4 * 4 + u * 4) + u(4 + u)) 20 + -13u + 2u2 = ((16 + 4u) + u(4 + u)) 20 + -13u + 2u2 = (16 + 4u + (4 * u + u * u)) 20 + -13u + 2u2 = (16 + 4u + (4u + u2)) Combine like terms: 4u + 4u = 8u 20 + -13u + 2u2 = (16 + 8u + u2) Solving 20 + -13u + 2u2 = 16 + 8u + u2 Solving for variable 'u'. Reorder the terms: 20 + -16 + -13u + -8u + 2u2 + -1u2 = 16 + 8u + u2 + -16 + -8u + -1u2 Combine like terms: 20 + -16 = 4 4 + -13u + -8u + 2u2 + -1u2 = 16 + 8u + u2 + -16 + -8u + -1u2 Combine like terms: -13u + -8u = -21u 4 + -21u + 2u2 + -1u2 = 16 + 8u + u2 + -16 + -8u + -1u2 Combine like terms: 2u2 + -1u2 = 1u2 4 + -21u + 1u2 = 16 + 8u + u2 + -16 + -8u + -1u2 Reorder the terms: 4 + -21u + 1u2 = 16 + -16 + 8u + -8u + u2 + -1u2 Combine like terms: 16 + -16 = 0 4 + -21u + 1u2 = 0 + 8u + -8u + u2 + -1u2 4 + -21u + 1u2 = 8u + -8u + u2 + -1u2 Combine like terms: 8u + -8u = 0 4 + -21u + 1u2 = 0 + u2 + -1u2 4 + -21u + 1u2 = u2 + -1u2 Combine like terms: u2 + -1u2 = 0 4 + -21u + 1u2 = 0 Begin completing the square. Move the constant term to the right: Add '-4' to each side of the equation. 4 + -21u + -4 + u2 = 0 + -4 Reorder the terms: 4 + -4 + -21u + u2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -21u + u2 = 0 + -4 -21u + u2 = 0 + -4 Combine like terms: 0 + -4 = -4 -21u + u2 = -4 The u term is -21u. Take half its coefficient (-10.5). Square it (110.25) and add it to both sides. Add '110.25' to each side of the equation. -21u + 110.25 + u2 = -4 + 110.25 Reorder the terms: 110.25 + -21u + u2 = -4 + 110.25 Combine like terms: -4 + 110.25 = 106.25 110.25 + -21u + u2 = 106.25 Factor a perfect square on the left side: (u + -10.5)(u + -10.5) = 106.25 Calculate the square root of the right side: 10.307764064 Break this problem into two subproblems by setting (u + -10.5) equal to 10.307764064 and -10.307764064.Subproblem 1
u + -10.5 = 10.307764064 Simplifying u + -10.5 = 10.307764064 Reorder the terms: -10.5 + u = 10.307764064 Solving -10.5 + u = 10.307764064 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + u = 10.307764064 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + u = 10.307764064 + 10.5 u = 10.307764064 + 10.5 Combine like terms: 10.307764064 + 10.5 = 20.807764064 u = 20.807764064 Simplifying u = 20.807764064Subproblem 2
u + -10.5 = -10.307764064 Simplifying u + -10.5 = -10.307764064 Reorder the terms: -10.5 + u = -10.307764064 Solving -10.5 + u = -10.307764064 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + u = -10.307764064 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + u = -10.307764064 + 10.5 u = -10.307764064 + 10.5 Combine like terms: -10.307764064 + 10.5 = 0.192235936 u = 0.192235936 Simplifying u = 0.192235936Solution
The solution to the problem is based on the solutions from the subproblems. u = {20.807764064, 0.192235936}
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