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Simplifying (T + -5)(T + 4) = 54 Reorder the terms: (-5 + T)(T + 4) = 54 Reorder the terms: (-5 + T)(4 + T) = 54 Multiply (-5 + T) * (4 + T) (-5(4 + T) + T(4 + T)) = 54 ((4 * -5 + T * -5) + T(4 + T)) = 54 ((-20 + -5T) + T(4 + T)) = 54 (-20 + -5T + (4 * T + T * T)) = 54 (-20 + -5T + (4T + T2)) = 54 Combine like terms: -5T + 4T = -1T (-20 + -1T + T2) = 54 Solving -20 + -1T + T2 = 54 Solving for variable 'T'. Reorder the terms: -20 + -54 + -1T + T2 = 54 + -54 Combine like terms: -20 + -54 = -74 -74 + -1T + T2 = 54 + -54 Combine like terms: 54 + -54 = 0 -74 + -1T + T2 = 0 Begin completing the square. Move the constant term to the right: Add '74' to each side of the equation. -74 + -1T + 74 + T2 = 0 + 74 Reorder the terms: -74 + 74 + -1T + T2 = 0 + 74 Combine like terms: -74 + 74 = 0 0 + -1T + T2 = 0 + 74 -1T + T2 = 0 + 74 Combine like terms: 0 + 74 = 74 -1T + T2 = 74 The T term is -1T. 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. -1T + 0.25 + T2 = 74 + 0.25 Reorder the terms: 0.25 + -1T + T2 = 74 + 0.25 Combine like terms: 74 + 0.25 = 74.25 0.25 + -1T + T2 = 74.25 Factor a perfect square on the left side: (T + -0.5)(T + -0.5) = 74.25 Calculate the square root of the right side: 8.61684397 Break this problem into two subproblems by setting (T + -0.5) equal to 8.61684397 and -8.61684397.Subproblem 1
T + -0.5 = 8.61684397 Simplifying T + -0.5 = 8.61684397 Reorder the terms: -0.5 + T = 8.61684397 Solving -0.5 + T = 8.61684397 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + T = 8.61684397 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + T = 8.61684397 + 0.5 T = 8.61684397 + 0.5 Combine like terms: 8.61684397 + 0.5 = 9.11684397 T = 9.11684397 Simplifying T = 9.11684397Subproblem 2
T + -0.5 = -8.61684397 Simplifying T + -0.5 = -8.61684397 Reorder the terms: -0.5 + T = -8.61684397 Solving -0.5 + T = -8.61684397 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + T = -8.61684397 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + T = -8.61684397 + 0.5 T = -8.61684397 + 0.5 Combine like terms: -8.61684397 + 0.5 = -8.11684397 T = -8.11684397 Simplifying T = -8.11684397Solution
The solution to the problem is based on the solutions from the subproblems. T = {9.11684397, -8.11684397}
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