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Simplifying -24t2 + 71t + 192 = 0 Reorder the terms: 192 + 71t + -24t2 = 0 Solving 192 + 71t + -24t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -24 the coefficient of the squared term: Divide each side by '-24'. -8 + -2.958333333t + t2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + -2.958333333t + 8 + t2 = 0 + 8 Reorder the terms: -8 + 8 + -2.958333333t + t2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + -2.958333333t + t2 = 0 + 8 -2.958333333t + t2 = 0 + 8 Combine like terms: 0 + 8 = 8 -2.958333333t + t2 = 8 The t term is -2.958333333t. Take half its coefficient (-1.479166667). Square it (2.187934029) and add it to both sides. Add '2.187934029' to each side of the equation. -2.958333333t + 2.187934029 + t2 = 8 + 2.187934029 Reorder the terms: 2.187934029 + -2.958333333t + t2 = 8 + 2.187934029 Combine like terms: 8 + 2.187934029 = 10.187934029 2.187934029 + -2.958333333t + t2 = 10.187934029 Factor a perfect square on the left side: (t + -1.479166667)(t + -1.479166667) = 10.187934029 Calculate the square root of the right side: 3.191854325 Break this problem into two subproblems by setting (t + -1.479166667) equal to 3.191854325 and -3.191854325.Subproblem 1
t + -1.479166667 = 3.191854325 Simplifying t + -1.479166667 = 3.191854325 Reorder the terms: -1.479166667 + t = 3.191854325 Solving -1.479166667 + t = 3.191854325 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.479166667' to each side of the equation. -1.479166667 + 1.479166667 + t = 3.191854325 + 1.479166667 Combine like terms: -1.479166667 + 1.479166667 = 0.000000000 0.000000000 + t = 3.191854325 + 1.479166667 t = 3.191854325 + 1.479166667 Combine like terms: 3.191854325 + 1.479166667 = 4.671020992 t = 4.671020992 Simplifying t = 4.671020992Subproblem 2
t + -1.479166667 = -3.191854325 Simplifying t + -1.479166667 = -3.191854325 Reorder the terms: -1.479166667 + t = -3.191854325 Solving -1.479166667 + t = -3.191854325 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.479166667' to each side of the equation. -1.479166667 + 1.479166667 + t = -3.191854325 + 1.479166667 Combine like terms: -1.479166667 + 1.479166667 = 0.000000000 0.000000000 + t = -3.191854325 + 1.479166667 t = -3.191854325 + 1.479166667 Combine like terms: -3.191854325 + 1.479166667 = -1.712687658 t = -1.712687658 Simplifying t = -1.712687658Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.671020992, -1.712687658}
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