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Simplifying t = 35 + 12t + -16t2 Solving t = 35 + 12t + -16t2 Solving for variable 't'. Reorder the terms: -35 + t + -12t + 16t2 = 35 + 12t + -16t2 + -35 + -12t + 16t2 Combine like terms: t + -12t = -11t -35 + -11t + 16t2 = 35 + 12t + -16t2 + -35 + -12t + 16t2 Reorder the terms: -35 + -11t + 16t2 = 35 + -35 + 12t + -12t + -16t2 + 16t2 Combine like terms: 35 + -35 = 0 -35 + -11t + 16t2 = 0 + 12t + -12t + -16t2 + 16t2 -35 + -11t + 16t2 = 12t + -12t + -16t2 + 16t2 Combine like terms: 12t + -12t = 0 -35 + -11t + 16t2 = 0 + -16t2 + 16t2 -35 + -11t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -35 + -11t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -2.1875 + -0.6875t + t2 = 0 Move the constant term to the right: Add '2.1875' to each side of the equation. -2.1875 + -0.6875t + 2.1875 + t2 = 0 + 2.1875 Reorder the terms: -2.1875 + 2.1875 + -0.6875t + t2 = 0 + 2.1875 Combine like terms: -2.1875 + 2.1875 = 0.0000 0.0000 + -0.6875t + t2 = 0 + 2.1875 -0.6875t + t2 = 0 + 2.1875 Combine like terms: 0 + 2.1875 = 2.1875 -0.6875t + t2 = 2.1875 The t term is -0.6875t. Take half its coefficient (-0.34375). Square it (0.1181640625) and add it to both sides. Add '0.1181640625' to each side of the equation. -0.6875t + 0.1181640625 + t2 = 2.1875 + 0.1181640625 Reorder the terms: 0.1181640625 + -0.6875t + t2 = 2.1875 + 0.1181640625 Combine like terms: 2.1875 + 0.1181640625 = 2.3056640625 0.1181640625 + -0.6875t + t2 = 2.3056640625 Factor a perfect square on the left side: (t + -0.34375)(t + -0.34375) = 2.3056640625 Calculate the square root of the right side: 1.518441327 Break this problem into two subproblems by setting (t + -0.34375) equal to 1.518441327 and -1.518441327.Subproblem 1
t + -0.34375 = 1.518441327 Simplifying t + -0.34375 = 1.518441327 Reorder the terms: -0.34375 + t = 1.518441327 Solving -0.34375 + t = 1.518441327 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.34375' to each side of the equation. -0.34375 + 0.34375 + t = 1.518441327 + 0.34375 Combine like terms: -0.34375 + 0.34375 = 0.00000 0.00000 + t = 1.518441327 + 0.34375 t = 1.518441327 + 0.34375 Combine like terms: 1.518441327 + 0.34375 = 1.862191327 t = 1.862191327 Simplifying t = 1.862191327Subproblem 2
t + -0.34375 = -1.518441327 Simplifying t + -0.34375 = -1.518441327 Reorder the terms: -0.34375 + t = -1.518441327 Solving -0.34375 + t = -1.518441327 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.34375' to each side of the equation. -0.34375 + 0.34375 + t = -1.518441327 + 0.34375 Combine like terms: -0.34375 + 0.34375 = 0.00000 0.00000 + t = -1.518441327 + 0.34375 t = -1.518441327 + 0.34375 Combine like terms: -1.518441327 + 0.34375 = -1.174691327 t = -1.174691327 Simplifying t = -1.174691327Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.862191327, -1.174691327}
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