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Simplifying -16T2 + 128T + 32 = 0 Reorder the terms: 32 + 128T + -16T2 = 0 Solving 32 + 128T + -16T2 = 0 Solving for variable 'T'. Factor out the Greatest Common Factor (GCF), '16'. 16(2 + 8T + -1T2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(2 + 8T + -1T2)' equal to zero and attempt to solve: Simplifying 2 + 8T + -1T2 = 0 Solving 2 + 8T + -1T2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -2 + -8T + T2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -8T + 2 + T2 = 0 + 2 Reorder the terms: -2 + 2 + -8T + T2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -8T + T2 = 0 + 2 -8T + T2 = 0 + 2 Combine like terms: 0 + 2 = 2 -8T + T2 = 2 The T term is -8T. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8T + 16 + T2 = 2 + 16 Reorder the terms: 16 + -8T + T2 = 2 + 16 Combine like terms: 2 + 16 = 18 16 + -8T + T2 = 18 Factor a perfect square on the left side: (T + -4)(T + -4) = 18 Calculate the square root of the right side: 4.242640687 Break this problem into two subproblems by setting (T + -4) equal to 4.242640687 and -4.242640687.Subproblem 1
T + -4 = 4.242640687 Simplifying T + -4 = 4.242640687 Reorder the terms: -4 + T = 4.242640687 Solving -4 + T = 4.242640687 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + T = 4.242640687 + 4 Combine like terms: -4 + 4 = 0 0 + T = 4.242640687 + 4 T = 4.242640687 + 4 Combine like terms: 4.242640687 + 4 = 8.242640687 T = 8.242640687 Simplifying T = 8.242640687Subproblem 2
T + -4 = -4.242640687 Simplifying T + -4 = -4.242640687 Reorder the terms: -4 + T = -4.242640687 Solving -4 + T = -4.242640687 Solving for variable 'T'. Move all terms containing T to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + T = -4.242640687 + 4 Combine like terms: -4 + 4 = 0 0 + T = -4.242640687 + 4 T = -4.242640687 + 4 Combine like terms: -4.242640687 + 4 = -0.242640687 T = -0.242640687 Simplifying T = -0.242640687Solution
The solution to the problem is based on the solutions from the subproblems. T = {8.242640687, -0.242640687}Solution
T = {8.242640687, -0.242640687}
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