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Simplifying t = -16t2 + 96t + 20 Reorder the terms: t = 20 + 96t + -16t2 Solving t = 20 + 96t + -16t2 Solving for variable 't'. Reorder the terms: -20 + t + -96t + 16t2 = 20 + 96t + -16t2 + -20 + -96t + 16t2 Combine like terms: t + -96t = -95t -20 + -95t + 16t2 = 20 + 96t + -16t2 + -20 + -96t + 16t2 Reorder the terms: -20 + -95t + 16t2 = 20 + -20 + 96t + -96t + -16t2 + 16t2 Combine like terms: 20 + -20 = 0 -20 + -95t + 16t2 = 0 + 96t + -96t + -16t2 + 16t2 -20 + -95t + 16t2 = 96t + -96t + -16t2 + 16t2 Combine like terms: 96t + -96t = 0 -20 + -95t + 16t2 = 0 + -16t2 + 16t2 -20 + -95t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -20 + -95t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.25 + -5.9375t + t2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -5.9375t + 1.25 + t2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + -5.9375t + t2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -5.9375t + t2 = 0 + 1.25 -5.9375t + t2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 -5.9375t + t2 = 1.25 The t term is -5.9375t. Take half its coefficient (-2.96875). Square it (8.813476563) and add it to both sides. Add '8.813476563' to each side of the equation. -5.9375t + 8.813476563 + t2 = 1.25 + 8.813476563 Reorder the terms: 8.813476563 + -5.9375t + t2 = 1.25 + 8.813476563 Combine like terms: 1.25 + 8.813476563 = 10.063476563 8.813476563 + -5.9375t + t2 = 10.063476563 Factor a perfect square on the left side: (t + -2.96875)(t + -2.96875) = 10.063476563 Calculate the square root of the right side: 3.172298309 Break this problem into two subproblems by setting (t + -2.96875) equal to 3.172298309 and -3.172298309.Subproblem 1
t + -2.96875 = 3.172298309 Simplifying t + -2.96875 = 3.172298309 Reorder the terms: -2.96875 + t = 3.172298309 Solving -2.96875 + t = 3.172298309 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.96875' to each side of the equation. -2.96875 + 2.96875 + t = 3.172298309 + 2.96875 Combine like terms: -2.96875 + 2.96875 = 0.00000 0.00000 + t = 3.172298309 + 2.96875 t = 3.172298309 + 2.96875 Combine like terms: 3.172298309 + 2.96875 = 6.141048309 t = 6.141048309 Simplifying t = 6.141048309Subproblem 2
t + -2.96875 = -3.172298309 Simplifying t + -2.96875 = -3.172298309 Reorder the terms: -2.96875 + t = -3.172298309 Solving -2.96875 + t = -3.172298309 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.96875' to each side of the equation. -2.96875 + 2.96875 + t = -3.172298309 + 2.96875 Combine like terms: -2.96875 + 2.96875 = 0.00000 0.00000 + t = -3.172298309 + 2.96875 t = -3.172298309 + 2.96875 Combine like terms: -3.172298309 + 2.96875 = -0.203548309 t = -0.203548309 Simplifying t = -0.203548309Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.141048309, -0.203548309}
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