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Simplifying -16t2 + 45t + 3 = 20 Reorder the terms: 3 + 45t + -16t2 = 20 Solving 3 + 45t + -16t2 = 20 Solving for variable 't'. Reorder the terms: 3 + -20 + 45t + -16t2 = 20 + -20 Combine like terms: 3 + -20 = -17 -17 + 45t + -16t2 = 20 + -20 Combine like terms: 20 + -20 = 0 -17 + 45t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 1.0625 + -2.8125t + t2 = 0 Move the constant term to the right: Add '-1.0625' to each side of the equation. 1.0625 + -2.8125t + -1.0625 + t2 = 0 + -1.0625 Reorder the terms: 1.0625 + -1.0625 + -2.8125t + t2 = 0 + -1.0625 Combine like terms: 1.0625 + -1.0625 = 0.0000 0.0000 + -2.8125t + t2 = 0 + -1.0625 -2.8125t + t2 = 0 + -1.0625 Combine like terms: 0 + -1.0625 = -1.0625 -2.8125t + t2 = -1.0625 The t term is -2.8125t. Take half its coefficient (-1.40625). Square it (1.977539063) and add it to both sides. Add '1.977539063' to each side of the equation. -2.8125t + 1.977539063 + t2 = -1.0625 + 1.977539063 Reorder the terms: 1.977539063 + -2.8125t + t2 = -1.0625 + 1.977539063 Combine like terms: -1.0625 + 1.977539063 = 0.915039063 1.977539063 + -2.8125t + t2 = 0.915039063 Factor a perfect square on the left side: (t + -1.40625)(t + -1.40625) = 0.915039063 Calculate the square root of the right side: 0.956576742 Break this problem into two subproblems by setting (t + -1.40625) equal to 0.956576742 and -0.956576742.Subproblem 1
t + -1.40625 = 0.956576742 Simplifying t + -1.40625 = 0.956576742 Reorder the terms: -1.40625 + t = 0.956576742 Solving -1.40625 + t = 0.956576742 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.40625' to each side of the equation. -1.40625 + 1.40625 + t = 0.956576742 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = 0.956576742 + 1.40625 t = 0.956576742 + 1.40625 Combine like terms: 0.956576742 + 1.40625 = 2.362826742 t = 2.362826742 Simplifying t = 2.362826742Subproblem 2
t + -1.40625 = -0.956576742 Simplifying t + -1.40625 = -0.956576742 Reorder the terms: -1.40625 + t = -0.956576742 Solving -1.40625 + t = -0.956576742 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.40625' to each side of the equation. -1.40625 + 1.40625 + t = -0.956576742 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + t = -0.956576742 + 1.40625 t = -0.956576742 + 1.40625 Combine like terms: -0.956576742 + 1.40625 = 0.449673258 t = 0.449673258 Simplifying t = 0.449673258Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.362826742, 0.449673258}
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