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Simplifying -16t2 + 39t + 324 = 0 Reorder the terms: 324 + 39t + -16t2 = 0 Solving 324 + 39t + -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'. -20.25 + -2.4375t + t2 = 0 Move the constant term to the right: Add '20.25' to each side of the equation. -20.25 + -2.4375t + 20.25 + t2 = 0 + 20.25 Reorder the terms: -20.25 + 20.25 + -2.4375t + t2 = 0 + 20.25 Combine like terms: -20.25 + 20.25 = 0.00 0.00 + -2.4375t + t2 = 0 + 20.25 -2.4375t + t2 = 0 + 20.25 Combine like terms: 0 + 20.25 = 20.25 -2.4375t + t2 = 20.25 The t term is -2.4375t. Take half its coefficient (-1.21875). Square it (1.485351563) and add it to both sides. Add '1.485351563' to each side of the equation. -2.4375t + 1.485351563 + t2 = 20.25 + 1.485351563 Reorder the terms: 1.485351563 + -2.4375t + t2 = 20.25 + 1.485351563 Combine like terms: 20.25 + 1.485351563 = 21.735351563 1.485351563 + -2.4375t + t2 = 21.735351563 Factor a perfect square on the left side: (t + -1.21875)(t + -1.21875) = 21.735351563 Calculate the square root of the right side: 4.662118785 Break this problem into two subproblems by setting (t + -1.21875) equal to 4.662118785 and -4.662118785.Subproblem 1
t + -1.21875 = 4.662118785 Simplifying t + -1.21875 = 4.662118785 Reorder the terms: -1.21875 + t = 4.662118785 Solving -1.21875 + t = 4.662118785 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.21875' to each side of the equation. -1.21875 + 1.21875 + t = 4.662118785 + 1.21875 Combine like terms: -1.21875 + 1.21875 = 0.00000 0.00000 + t = 4.662118785 + 1.21875 t = 4.662118785 + 1.21875 Combine like terms: 4.662118785 + 1.21875 = 5.880868785 t = 5.880868785 Simplifying t = 5.880868785Subproblem 2
t + -1.21875 = -4.662118785 Simplifying t + -1.21875 = -4.662118785 Reorder the terms: -1.21875 + t = -4.662118785 Solving -1.21875 + t = -4.662118785 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.21875' to each side of the equation. -1.21875 + 1.21875 + t = -4.662118785 + 1.21875 Combine like terms: -1.21875 + 1.21875 = 0.00000 0.00000 + t = -4.662118785 + 1.21875 t = -4.662118785 + 1.21875 Combine like terms: -4.662118785 + 1.21875 = -3.443368785 t = -3.443368785 Simplifying t = -3.443368785Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.880868785, -3.443368785}
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