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Simplifying 0 = 16t2 + 44t + -24 Reorder the terms: 0 = -24 + 44t + 16t2 Solving 0 = -24 + 44t + 16t2 Solving for variable 't'. Combine like terms: 0 + 24 = 24 24 + -44t + -16t2 = -24 + 44t + 16t2 + 24 + -44t + -16t2 Reorder the terms: 24 + -44t + -16t2 = -24 + 24 + 44t + -44t + 16t2 + -16t2 Combine like terms: -24 + 24 = 0 24 + -44t + -16t2 = 0 + 44t + -44t + 16t2 + -16t2 24 + -44t + -16t2 = 44t + -44t + 16t2 + -16t2 Combine like terms: 44t + -44t = 0 24 + -44t + -16t2 = 0 + 16t2 + -16t2 24 + -44t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 24 + -44t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(6 + -11t + -4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(6 + -11t + -4t2)' equal to zero and attempt to solve: Simplifying 6 + -11t + -4t2 = 0 Solving 6 + -11t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1.5 + 2.75t + t2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 2.75t + 1.5 + t2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 2.75t + t2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 2.75t + t2 = 0 + 1.5 2.75t + t2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 2.75t + t2 = 1.5 The t term is 2.75t. Take half its coefficient (1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. 2.75t + 1.890625 + t2 = 1.5 + 1.890625 Reorder the terms: 1.890625 + 2.75t + t2 = 1.5 + 1.890625 Combine like terms: 1.5 + 1.890625 = 3.390625 1.890625 + 2.75t + t2 = 3.390625 Factor a perfect square on the left side: (t + 1.375)(t + 1.375) = 3.390625 Calculate the square root of the right side: 1.841364983 Break this problem into two subproblems by setting (t + 1.375) equal to 1.841364983 and -1.841364983.Subproblem 1
t + 1.375 = 1.841364983 Simplifying t + 1.375 = 1.841364983 Reorder the terms: 1.375 + t = 1.841364983 Solving 1.375 + t = 1.841364983 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + t = 1.841364983 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + t = 1.841364983 + -1.375 t = 1.841364983 + -1.375 Combine like terms: 1.841364983 + -1.375 = 0.466364983 t = 0.466364983 Simplifying t = 0.466364983Subproblem 2
t + 1.375 = -1.841364983 Simplifying t + 1.375 = -1.841364983 Reorder the terms: 1.375 + t = -1.841364983 Solving 1.375 + t = -1.841364983 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + t = -1.841364983 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + t = -1.841364983 + -1.375 t = -1.841364983 + -1.375 Combine like terms: -1.841364983 + -1.375 = -3.216364983 t = -3.216364983 Simplifying t = -3.216364983Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.466364983, -3.216364983}Solution
t = {0.466364983, -3.216364983}
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