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Simplifying -16t2 + 112t + 128 = 192 Reorder the terms: 128 + 112t + -16t2 = 192 Solving 128 + 112t + -16t2 = 192 Solving for variable 't'. Reorder the terms: 128 + -192 + 112t + -16t2 = 192 + -192 Combine like terms: 128 + -192 = -64 -64 + 112t + -16t2 = 192 + -192 Combine like terms: 192 + -192 = 0 -64 + 112t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '16'. 16(-4 + 7t + -1t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(-4 + 7t + -1t2)' equal to zero and attempt to solve: Simplifying -4 + 7t + -1t2 = 0 Solving -4 + 7t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 4 + -7t + t2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + -7t + -4 + t2 = 0 + -4 Reorder the terms: 4 + -4 + -7t + t2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -7t + t2 = 0 + -4 -7t + t2 = 0 + -4 Combine like terms: 0 + -4 = -4 -7t + t2 = -4 The t term is -7t. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7t + 12.25 + t2 = -4 + 12.25 Reorder the terms: 12.25 + -7t + t2 = -4 + 12.25 Combine like terms: -4 + 12.25 = 8.25 12.25 + -7t + t2 = 8.25 Factor a perfect square on the left side: (t + -3.5)(t + -3.5) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (t + -3.5) equal to 2.872281323 and -2.872281323.Subproblem 1
t + -3.5 = 2.872281323 Simplifying t + -3.5 = 2.872281323 Reorder the terms: -3.5 + t = 2.872281323 Solving -3.5 + t = 2.872281323 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + t = 2.872281323 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + t = 2.872281323 + 3.5 t = 2.872281323 + 3.5 Combine like terms: 2.872281323 + 3.5 = 6.372281323 t = 6.372281323 Simplifying t = 6.372281323Subproblem 2
t + -3.5 = -2.872281323 Simplifying t + -3.5 = -2.872281323 Reorder the terms: -3.5 + t = -2.872281323 Solving -3.5 + t = -2.872281323 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + t = -2.872281323 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + t = -2.872281323 + 3.5 t = -2.872281323 + 3.5 Combine like terms: -2.872281323 + 3.5 = 0.627718677 t = 0.627718677 Simplifying t = 0.627718677Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.372281323, 0.627718677}Solution
t = {6.372281323, 0.627718677}
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