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Simplifying -16t2 + 320t = 1000 Reorder the terms: 320t + -16t2 = 1000 Solving 320t + -16t2 = 1000 Solving for variable 't'. Reorder the terms: -1000 + 320t + -16t2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -1000 + 320t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-125 + 40t + -2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-125 + 40t + -2t2)' equal to zero and attempt to solve: Simplifying -125 + 40t + -2t2 = 0 Solving -125 + 40t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 62.5 + -20t + t2 = 0 Move the constant term to the right: Add '-62.5' to each side of the equation. 62.5 + -20t + -62.5 + t2 = 0 + -62.5 Reorder the terms: 62.5 + -62.5 + -20t + t2 = 0 + -62.5 Combine like terms: 62.5 + -62.5 = 0.0 0.0 + -20t + t2 = 0 + -62.5 -20t + t2 = 0 + -62.5 Combine like terms: 0 + -62.5 = -62.5 -20t + t2 = -62.5 The t term is -20t. Take half its coefficient (-10). Square it (100) and add it to both sides. Add '100' to each side of the equation. -20t + 100 + t2 = -62.5 + 100 Reorder the terms: 100 + -20t + t2 = -62.5 + 100 Combine like terms: -62.5 + 100 = 37.5 100 + -20t + t2 = 37.5 Factor a perfect square on the left side: (t + -10)(t + -10) = 37.5 Calculate the square root of the right side: 6.123724357 Break this problem into two subproblems by setting (t + -10) equal to 6.123724357 and -6.123724357.Subproblem 1
t + -10 = 6.123724357 Simplifying t + -10 = 6.123724357 Reorder the terms: -10 + t = 6.123724357 Solving -10 + t = 6.123724357 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + t = 6.123724357 + 10 Combine like terms: -10 + 10 = 0 0 + t = 6.123724357 + 10 t = 6.123724357 + 10 Combine like terms: 6.123724357 + 10 = 16.123724357 t = 16.123724357 Simplifying t = 16.123724357Subproblem 2
t + -10 = -6.123724357 Simplifying t + -10 = -6.123724357 Reorder the terms: -10 + t = -6.123724357 Solving -10 + t = -6.123724357 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + t = -6.123724357 + 10 Combine like terms: -10 + 10 = 0 0 + t = -6.123724357 + 10 t = -6.123724357 + 10 Combine like terms: -6.123724357 + 10 = 3.876275643 t = 3.876275643 Simplifying t = 3.876275643Solution
The solution to the problem is based on the solutions from the subproblems. t = {16.123724357, 3.876275643}Solution
t = {16.123724357, 3.876275643}
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