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Simplifying 0 = -16t2 + 72t + 3 Reorder the terms: 0 = 3 + 72t + -16t2 Solving 0 = 3 + 72t + -16t2 Solving for variable 't'. Combine like terms: 0 + -3 = -3 -3 + -72t + 16t2 = 3 + 72t + -16t2 + -3 + -72t + 16t2 Reorder the terms: -3 + -72t + 16t2 = 3 + -3 + 72t + -72t + -16t2 + 16t2 Combine like terms: 3 + -3 = 0 -3 + -72t + 16t2 = 0 + 72t + -72t + -16t2 + 16t2 -3 + -72t + 16t2 = 72t + -72t + -16t2 + 16t2 Combine like terms: 72t + -72t = 0 -3 + -72t + 16t2 = 0 + -16t2 + 16t2 -3 + -72t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -3 + -72t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.1875 + -4.5t + t2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -4.5t + 0.1875 + t2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -4.5t + t2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -4.5t + t2 = 0 + 0.1875 -4.5t + t2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 -4.5t + t2 = 0.1875 The t term is -4.5t. Take half its coefficient (-2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. -4.5t + 5.0625 + t2 = 0.1875 + 5.0625 Reorder the terms: 5.0625 + -4.5t + t2 = 0.1875 + 5.0625 Combine like terms: 0.1875 + 5.0625 = 5.25 5.0625 + -4.5t + t2 = 5.25 Factor a perfect square on the left side: (t + -2.25)(t + -2.25) = 5.25 Calculate the square root of the right side: 2.291287847 Break this problem into two subproblems by setting (t + -2.25) equal to 2.291287847 and -2.291287847.Subproblem 1
t + -2.25 = 2.291287847 Simplifying t + -2.25 = 2.291287847 Reorder the terms: -2.25 + t = 2.291287847 Solving -2.25 + t = 2.291287847 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + t = 2.291287847 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + t = 2.291287847 + 2.25 t = 2.291287847 + 2.25 Combine like terms: 2.291287847 + 2.25 = 4.541287847 t = 4.541287847 Simplifying t = 4.541287847Subproblem 2
t + -2.25 = -2.291287847 Simplifying t + -2.25 = -2.291287847 Reorder the terms: -2.25 + t = -2.291287847 Solving -2.25 + t = -2.291287847 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + t = -2.291287847 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + t = -2.291287847 + 2.25 t = -2.291287847 + 2.25 Combine like terms: -2.291287847 + 2.25 = -0.041287847 t = -0.041287847 Simplifying t = -0.041287847Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.541287847, -0.041287847}
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