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Simplifying t = -16t2 + 128t + 20 Reorder the terms: t = 20 + 128t + -16t2 Solving t = 20 + 128t + -16t2 Solving for variable 't'. Reorder the terms: -20 + t + -128t + 16t2 = 20 + 128t + -16t2 + -20 + -128t + 16t2 Combine like terms: t + -128t = -127t -20 + -127t + 16t2 = 20 + 128t + -16t2 + -20 + -128t + 16t2 Reorder the terms: -20 + -127t + 16t2 = 20 + -20 + 128t + -128t + -16t2 + 16t2 Combine like terms: 20 + -20 = 0 -20 + -127t + 16t2 = 0 + 128t + -128t + -16t2 + 16t2 -20 + -127t + 16t2 = 128t + -128t + -16t2 + 16t2 Combine like terms: 128t + -128t = 0 -20 + -127t + 16t2 = 0 + -16t2 + 16t2 -20 + -127t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -20 + -127t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.25 + -7.9375t + t2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -7.9375t + 1.25 + t2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + -7.9375t + t2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -7.9375t + t2 = 0 + 1.25 -7.9375t + t2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 -7.9375t + t2 = 1.25 The t term is -7.9375t. Take half its coefficient (-3.96875). Square it (15.75097656) and add it to both sides. Add '15.75097656' to each side of the equation. -7.9375t + 15.75097656 + t2 = 1.25 + 15.75097656 Reorder the terms: 15.75097656 + -7.9375t + t2 = 1.25 + 15.75097656 Combine like terms: 1.25 + 15.75097656 = 17.00097656 15.75097656 + -7.9375t + t2 = 17.00097656 Factor a perfect square on the left side: (t + -3.96875)(t + -3.96875) = 17.00097656 Calculate the square root of the right side: 4.123224049 Break this problem into two subproblems by setting (t + -3.96875) equal to 4.123224049 and -4.123224049.Subproblem 1
t + -3.96875 = 4.123224049 Simplifying t + -3.96875 = 4.123224049 Reorder the terms: -3.96875 + t = 4.123224049 Solving -3.96875 + t = 4.123224049 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.96875' to each side of the equation. -3.96875 + 3.96875 + t = 4.123224049 + 3.96875 Combine like terms: -3.96875 + 3.96875 = 0.00000 0.00000 + t = 4.123224049 + 3.96875 t = 4.123224049 + 3.96875 Combine like terms: 4.123224049 + 3.96875 = 8.091974049 t = 8.091974049 Simplifying t = 8.091974049Subproblem 2
t + -3.96875 = -4.123224049 Simplifying t + -3.96875 = -4.123224049 Reorder the terms: -3.96875 + t = -4.123224049 Solving -3.96875 + t = -4.123224049 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.96875' to each side of the equation. -3.96875 + 3.96875 + t = -4.123224049 + 3.96875 Combine like terms: -3.96875 + 3.96875 = 0.00000 0.00000 + t = -4.123224049 + 3.96875 t = -4.123224049 + 3.96875 Combine like terms: -4.123224049 + 3.96875 = -0.154474049 t = -0.154474049 Simplifying t = -0.154474049Solution
The solution to the problem is based on the solutions from the subproblems. t = {8.091974049, -0.154474049}
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