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Simplifying 24 = 72t2 + -16t Reorder the terms: 24 = -16t + 72t2 Solving 24 = -16t + 72t2 Solving for variable 't'. Reorder the terms: 24 + 16t + -72t2 = -16t + 16t + 72t2 + -72t2 Combine like terms: -16t + 16t = 0 24 + 16t + -72t2 = 0 + 72t2 + -72t2 24 + 16t + -72t2 = 72t2 + -72t2 Combine like terms: 72t2 + -72t2 = 0 24 + 16t + -72t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(3 + 2t + -9t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(3 + 2t + -9t2)' equal to zero and attempt to solve: Simplifying 3 + 2t + -9t2 = 0 Solving 3 + 2t + -9t2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -0.3333333333 + -0.2222222222t + t2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -0.2222222222t + 0.3333333333 + t2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -0.2222222222t + t2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -0.2222222222t + t2 = 0 + 0.3333333333 -0.2222222222t + t2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -0.2222222222t + t2 = 0.3333333333 The t term is -0.2222222222t. Take half its coefficient (-0.1111111111). Square it (0.01234567901) and add it to both sides. Add '0.01234567901' to each side of the equation. -0.2222222222t + 0.01234567901 + t2 = 0.3333333333 + 0.01234567901 Reorder the terms: 0.01234567901 + -0.2222222222t + t2 = 0.3333333333 + 0.01234567901 Combine like terms: 0.3333333333 + 0.01234567901 = 0.34567901231 0.01234567901 + -0.2222222222t + t2 = 0.34567901231 Factor a perfect square on the left side: (t + -0.1111111111)(t + -0.1111111111) = 0.34567901231 Calculate the square root of the right side: 0.587944736 Break this problem into two subproblems by setting (t + -0.1111111111) equal to 0.587944736 and -0.587944736.Subproblem 1
t + -0.1111111111 = 0.587944736 Simplifying t + -0.1111111111 = 0.587944736 Reorder the terms: -0.1111111111 + t = 0.587944736 Solving -0.1111111111 + t = 0.587944736 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1111111111' to each side of the equation. -0.1111111111 + 0.1111111111 + t = 0.587944736 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + t = 0.587944736 + 0.1111111111 t = 0.587944736 + 0.1111111111 Combine like terms: 0.587944736 + 0.1111111111 = 0.6990558471 t = 0.6990558471 Simplifying t = 0.6990558471Subproblem 2
t + -0.1111111111 = -0.587944736 Simplifying t + -0.1111111111 = -0.587944736 Reorder the terms: -0.1111111111 + t = -0.587944736 Solving -0.1111111111 + t = -0.587944736 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1111111111' to each side of the equation. -0.1111111111 + 0.1111111111 + t = -0.587944736 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + t = -0.587944736 + 0.1111111111 t = -0.587944736 + 0.1111111111 Combine like terms: -0.587944736 + 0.1111111111 = -0.4768336249 t = -0.4768336249 Simplifying t = -0.4768336249Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.6990558471, -0.4768336249}Solution
t = {0.6990558471, -0.4768336249}
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