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Simplifying -16t2 + 40t + 65 = 0 Reorder the terms: 65 + 40t + -16t2 = 0 Solving 65 + 40t + -16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -4.0625 + -2.5t + t2 = 0 Move the constant term to the right: Add '4.0625' to each side of the equation. -4.0625 + -2.5t + 4.0625 + t2 = 0 + 4.0625 Reorder the terms: -4.0625 + 4.0625 + -2.5t + t2 = 0 + 4.0625 Combine like terms: -4.0625 + 4.0625 = 0.0000 0.0000 + -2.5t + t2 = 0 + 4.0625 -2.5t + t2 = 0 + 4.0625 Combine like terms: 0 + 4.0625 = 4.0625 -2.5t + t2 = 4.0625 The t term is -2.5t. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5t + 1.5625 + t2 = 4.0625 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = 4.0625 + 1.5625 Combine like terms: 4.0625 + 1.5625 = 5.625 1.5625 + -2.5t + t2 = 5.625 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 5.625 Calculate the square root of the right side: 2.371708245 Break this problem into two subproblems by setting (t + -1.25) equal to 2.371708245 and -2.371708245.Subproblem 1
t + -1.25 = 2.371708245 Simplifying t + -1.25 = 2.371708245 Reorder the terms: -1.25 + t = 2.371708245 Solving -1.25 + t = 2.371708245 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = 2.371708245 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 2.371708245 + 1.25 t = 2.371708245 + 1.25 Combine like terms: 2.371708245 + 1.25 = 3.621708245 t = 3.621708245 Simplifying t = 3.621708245Subproblem 2
t + -1.25 = -2.371708245 Simplifying t + -1.25 = -2.371708245 Reorder the terms: -1.25 + t = -2.371708245 Solving -1.25 + t = -2.371708245 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = -2.371708245 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -2.371708245 + 1.25 t = -2.371708245 + 1.25 Combine like terms: -2.371708245 + 1.25 = -1.121708245 t = -1.121708245 Simplifying t = -1.121708245Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.621708245, -1.121708245}
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