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Simplifying t = -16t2 + 64t + 80 Reorder the terms: t = 80 + 64t + -16t2 Solving t = 80 + 64t + -16t2 Solving for variable 't'. Reorder the terms: -80 + t + -64t + 16t2 = 80 + 64t + -16t2 + -80 + -64t + 16t2 Combine like terms: t + -64t = -63t -80 + -63t + 16t2 = 80 + 64t + -16t2 + -80 + -64t + 16t2 Reorder the terms: -80 + -63t + 16t2 = 80 + -80 + 64t + -64t + -16t2 + 16t2 Combine like terms: 80 + -80 = 0 -80 + -63t + 16t2 = 0 + 64t + -64t + -16t2 + 16t2 -80 + -63t + 16t2 = 64t + -64t + -16t2 + 16t2 Combine like terms: 64t + -64t = 0 -80 + -63t + 16t2 = 0 + -16t2 + 16t2 -80 + -63t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -80 + -63t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -5 + -3.9375t + t2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -3.9375t + 5 + t2 = 0 + 5 Reorder the terms: -5 + 5 + -3.9375t + t2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -3.9375t + t2 = 0 + 5 -3.9375t + t2 = 0 + 5 Combine like terms: 0 + 5 = 5 -3.9375t + t2 = 5 The t term is -3.9375t. Take half its coefficient (-1.96875). Square it (3.875976563) and add it to both sides. Add '3.875976563' to each side of the equation. -3.9375t + 3.875976563 + t2 = 5 + 3.875976563 Reorder the terms: 3.875976563 + -3.9375t + t2 = 5 + 3.875976563 Combine like terms: 5 + 3.875976563 = 8.875976563 3.875976563 + -3.9375t + t2 = 8.875976563 Factor a perfect square on the left side: (t + -1.96875)(t + -1.96875) = 8.875976563 Calculate the square root of the right side: 2.97925772 Break this problem into two subproblems by setting (t + -1.96875) equal to 2.97925772 and -2.97925772.Subproblem 1
t + -1.96875 = 2.97925772 Simplifying t + -1.96875 = 2.97925772 Reorder the terms: -1.96875 + t = 2.97925772 Solving -1.96875 + t = 2.97925772 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.96875' to each side of the equation. -1.96875 + 1.96875 + t = 2.97925772 + 1.96875 Combine like terms: -1.96875 + 1.96875 = 0.00000 0.00000 + t = 2.97925772 + 1.96875 t = 2.97925772 + 1.96875 Combine like terms: 2.97925772 + 1.96875 = 4.94800772 t = 4.94800772 Simplifying t = 4.94800772Subproblem 2
t + -1.96875 = -2.97925772 Simplifying t + -1.96875 = -2.97925772 Reorder the terms: -1.96875 + t = -2.97925772 Solving -1.96875 + t = -2.97925772 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.96875' to each side of the equation. -1.96875 + 1.96875 + t = -2.97925772 + 1.96875 Combine like terms: -1.96875 + 1.96875 = 0.00000 0.00000 + t = -2.97925772 + 1.96875 t = -2.97925772 + 1.96875 Combine like terms: -2.97925772 + 1.96875 = -1.01050772 t = -1.01050772 Simplifying t = -1.01050772Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.94800772, -1.01050772}
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