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Simplifying -16t2 + 12t + 975 = 0 Reorder the terms: 975 + 12t + -16t2 = 0 Solving 975 + 12t + -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'. -60.9375 + -0.75t + t2 = 0 Move the constant term to the right: Add '60.9375' to each side of the equation. -60.9375 + -0.75t + 60.9375 + t2 = 0 + 60.9375 Reorder the terms: -60.9375 + 60.9375 + -0.75t + t2 = 0 + 60.9375 Combine like terms: -60.9375 + 60.9375 = 0.0000 0.0000 + -0.75t + t2 = 0 + 60.9375 -0.75t + t2 = 0 + 60.9375 Combine like terms: 0 + 60.9375 = 60.9375 -0.75t + t2 = 60.9375 The t term is -0.75t. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75t + 0.140625 + t2 = 60.9375 + 0.140625 Reorder the terms: 0.140625 + -0.75t + t2 = 60.9375 + 0.140625 Combine like terms: 60.9375 + 0.140625 = 61.078125 0.140625 + -0.75t + t2 = 61.078125 Factor a perfect square on the left side: (t + -0.375)(t + -0.375) = 61.078125 Calculate the square root of the right side: 7.815249516 Break this problem into two subproblems by setting (t + -0.375) equal to 7.815249516 and -7.815249516.Subproblem 1
t + -0.375 = 7.815249516 Simplifying t + -0.375 = 7.815249516 Reorder the terms: -0.375 + t = 7.815249516 Solving -0.375 + t = 7.815249516 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + t = 7.815249516 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = 7.815249516 + 0.375 t = 7.815249516 + 0.375 Combine like terms: 7.815249516 + 0.375 = 8.190249516 t = 8.190249516 Simplifying t = 8.190249516Subproblem 2
t + -0.375 = -7.815249516 Simplifying t + -0.375 = -7.815249516 Reorder the terms: -0.375 + t = -7.815249516 Solving -0.375 + t = -7.815249516 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + t = -7.815249516 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + t = -7.815249516 + 0.375 t = -7.815249516 + 0.375 Combine like terms: -7.815249516 + 0.375 = -7.440249516 t = -7.440249516 Simplifying t = -7.440249516Solution
The solution to the problem is based on the solutions from the subproblems. t = {8.190249516, -7.440249516}
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