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Simplifying T(2T + -3) = 30 Reorder the terms: T(-3 + 2T) = 30 (-3 * T + 2T * T) = 30 (-3T + 2T2) = 30 Solving -3T + 2T2 = 30 Solving for variable 'T'. Reorder the terms: -30 + -3T + 2T2 = 30 + -30 Combine like terms: 30 + -30 = 0 -30 + -3T + 2T2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -15 + -1.5T + T2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + -1.5T + 15 + T2 = 0 + 15 Reorder the terms: -15 + 15 + -1.5T + T2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + -1.5T + T2 = 0 + 15 -1.5T + T2 = 0 + 15 Combine like terms: 0 + 15 = 15 -1.5T + T2 = 15 The T term is -1.5T. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5T + 0.5625 + T2 = 15 + 0.5625 Reorder the terms: 0.5625 + -1.5T + T2 = 15 + 0.5625 Combine like terms: 15 + 0.5625 = 15.5625 0.5625 + -1.5T + T2 = 15.5625 Factor a perfect square on the left side: (T + -0.75)(T + -0.75) = 15.5625 Calculate the square root of the right side: 3.94493346 Break this problem into two subproblems by setting (T + -0.75) equal to 3.94493346 and -3.94493346.Subproblem 1
T + -0.75 = 3.94493346 Simplifying T + -0.75 = 3.94493346 Reorder the terms: -0.75 + T = 3.94493346 Solving -0.75 + T = 3.94493346 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + T = 3.94493346 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + T = 3.94493346 + 0.75 T = 3.94493346 + 0.75 Combine like terms: 3.94493346 + 0.75 = 4.69493346 T = 4.69493346 Simplifying T = 4.69493346Subproblem 2
T + -0.75 = -3.94493346 Simplifying T + -0.75 = -3.94493346 Reorder the terms: -0.75 + T = -3.94493346 Solving -0.75 + T = -3.94493346 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + T = -3.94493346 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + T = -3.94493346 + 0.75 T = -3.94493346 + 0.75 Combine like terms: -3.94493346 + 0.75 = -3.19493346 T = -3.19493346 Simplifying T = -3.19493346Solution
The solution to the problem is based on the solutions from the subproblems. T = {4.69493346, -3.19493346}
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