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Simplifying 120 = 40 + 80T + -5T2 Solving 120 = 40 + 80T + -5T2 Solving for variable 'T'. Combine like terms: 120 + -40 = 80 80 + -80T + 5T2 = 40 + 80T + -5T2 + -40 + -80T + 5T2 Reorder the terms: 80 + -80T + 5T2 = 40 + -40 + 80T + -80T + -5T2 + 5T2 Combine like terms: 40 + -40 = 0 80 + -80T + 5T2 = 0 + 80T + -80T + -5T2 + 5T2 80 + -80T + 5T2 = 80T + -80T + -5T2 + 5T2 Combine like terms: 80T + -80T = 0 80 + -80T + 5T2 = 0 + -5T2 + 5T2 80 + -80T + 5T2 = -5T2 + 5T2 Combine like terms: -5T2 + 5T2 = 0 80 + -80T + 5T2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(16 + -16T + T2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(16 + -16T + T2)' equal to zero and attempt to solve: Simplifying 16 + -16T + T2 = 0 Solving 16 + -16T + T2 = 0 Begin completing the square. Move the constant term to the right: Add '-16' to each side of the equation. 16 + -16T + -16 + T2 = 0 + -16 Reorder the terms: 16 + -16 + -16T + T2 = 0 + -16 Combine like terms: 16 + -16 = 0 0 + -16T + T2 = 0 + -16 -16T + T2 = 0 + -16 Combine like terms: 0 + -16 = -16 -16T + T2 = -16 The T term is -16T. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16T + 64 + T2 = -16 + 64 Reorder the terms: 64 + -16T + T2 = -16 + 64 Combine like terms: -16 + 64 = 48 64 + -16T + T2 = 48 Factor a perfect square on the left side: (T + -8)(T + -8) = 48 Calculate the square root of the right side: 6.92820323 Break this problem into two subproblems by setting (T + -8) equal to 6.92820323 and -6.92820323.Subproblem 1
T + -8 = 6.92820323 Simplifying T + -8 = 6.92820323 Reorder the terms: -8 + T = 6.92820323 Solving -8 + T = 6.92820323 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + T = 6.92820323 + 8 Combine like terms: -8 + 8 = 0 0 + T = 6.92820323 + 8 T = 6.92820323 + 8 Combine like terms: 6.92820323 + 8 = 14.92820323 T = 14.92820323 Simplifying T = 14.92820323Subproblem 2
T + -8 = -6.92820323 Simplifying T + -8 = -6.92820323 Reorder the terms: -8 + T = -6.92820323 Solving -8 + T = -6.92820323 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + T = -6.92820323 + 8 Combine like terms: -8 + 8 = 0 0 + T = -6.92820323 + 8 T = -6.92820323 + 8 Combine like terms: -6.92820323 + 8 = 1.07179677 T = 1.07179677 Simplifying T = 1.07179677Solution
The solution to the problem is based on the solutions from the subproblems. T = {14.92820323, 1.07179677}Solution
T = {14.92820323, 1.07179677}
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