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Simplifying 2.5 = -16T2 + 16T + 6.5 Reorder the terms: 2.5 = 6.5 + 16T + -16T2 Solving 2.5 = 6.5 + 16T + -16T2 Solving for variable 'T'. Combine like terms: 2.5 + -6.5 = -4 -4 + -16T + 16T2 = 6.5 + 16T + -16T2 + -6.5 + -16T + 16T2 Reorder the terms: -4 + -16T + 16T2 = 6.5 + -6.5 + 16T + -16T + -16T2 + 16T2 Combine like terms: 6.5 + -6.5 = 0.0 -4 + -16T + 16T2 = 0.0 + 16T + -16T + -16T2 + 16T2 -4 + -16T + 16T2 = 16T + -16T + -16T2 + 16T2 Combine like terms: 16T + -16T = 0 -4 + -16T + 16T2 = 0 + -16T2 + 16T2 -4 + -16T + 16T2 = -16T2 + 16T2 Combine like terms: -16T2 + 16T2 = 0 -4 + -16T + 16T2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-1 + -4T + 4T2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-1 + -4T + 4T2)' equal to zero and attempt to solve: Simplifying -1 + -4T + 4T2 = 0 Solving -1 + -4T + 4T2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + -1T + T2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -1T + 0.25 + T2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -1T + T2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -1T + T2 = 0 + 0.25 -1T + T2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -1T + T2 = 0.25 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 = 0.25 + 0.25 Reorder the terms: 0.25 + -1T + T2 = 0.25 + 0.25 Combine like terms: 0.25 + 0.25 = 0.5 0.25 + -1T + T2 = 0.5 Factor a perfect square on the left side: (T + -0.5)(T + -0.5) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (T + -0.5) equal to 0.707106781 and -0.707106781.Subproblem 1
T + -0.5 = 0.707106781 Simplifying T + -0.5 = 0.707106781 Reorder the terms: -0.5 + T = 0.707106781 Solving -0.5 + T = 0.707106781 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 = 0.707106781 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + T = 0.707106781 + 0.5 T = 0.707106781 + 0.5 Combine like terms: 0.707106781 + 0.5 = 1.207106781 T = 1.207106781 Simplifying T = 1.207106781Subproblem 2
T + -0.5 = -0.707106781 Simplifying T + -0.5 = -0.707106781 Reorder the terms: -0.5 + T = -0.707106781 Solving -0.5 + T = -0.707106781 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 = -0.707106781 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + T = -0.707106781 + 0.5 T = -0.707106781 + 0.5 Combine like terms: -0.707106781 + 0.5 = -0.207106781 T = -0.207106781 Simplifying T = -0.207106781Solution
The solution to the problem is based on the solutions from the subproblems. T = {1.207106781, -0.207106781}Solution
T = {1.207106781, -0.207106781}
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