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Simplifying -16t2 + 96t + 80 = 125 Reorder the terms: 80 + 96t + -16t2 = 125 Solving 80 + 96t + -16t2 = 125 Solving for variable 't'. Reorder the terms: 80 + -125 + 96t + -16t2 = 125 + -125 Combine like terms: 80 + -125 = -45 -45 + 96t + -16t2 = 125 + -125 Combine like terms: 125 + -125 = 0 -45 + 96t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 2.8125 + -6t + t2 = 0 Move the constant term to the right: Add '-2.8125' to each side of the equation. 2.8125 + -6t + -2.8125 + t2 = 0 + -2.8125 Reorder the terms: 2.8125 + -2.8125 + -6t + t2 = 0 + -2.8125 Combine like terms: 2.8125 + -2.8125 = 0.0000 0.0000 + -6t + t2 = 0 + -2.8125 -6t + t2 = 0 + -2.8125 Combine like terms: 0 + -2.8125 = -2.8125 -6t + t2 = -2.8125 The t term is -6t. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6t + 9 + t2 = -2.8125 + 9 Reorder the terms: 9 + -6t + t2 = -2.8125 + 9 Combine like terms: -2.8125 + 9 = 6.1875 9 + -6t + t2 = 6.1875 Factor a perfect square on the left side: (t + -3)(t + -3) = 6.1875 Calculate the square root of the right side: 2.487468593 Break this problem into two subproblems by setting (t + -3) equal to 2.487468593 and -2.487468593.Subproblem 1
t + -3 = 2.487468593 Simplifying t + -3 = 2.487468593 Reorder the terms: -3 + t = 2.487468593 Solving -3 + t = 2.487468593 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = 2.487468593 + 3 Combine like terms: -3 + 3 = 0 0 + t = 2.487468593 + 3 t = 2.487468593 + 3 Combine like terms: 2.487468593 + 3 = 5.487468593 t = 5.487468593 Simplifying t = 5.487468593Subproblem 2
t + -3 = -2.487468593 Simplifying t + -3 = -2.487468593 Reorder the terms: -3 + t = -2.487468593 Solving -3 + t = -2.487468593 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = -2.487468593 + 3 Combine like terms: -3 + 3 = 0 0 + t = -2.487468593 + 3 t = -2.487468593 + 3 Combine like terms: -2.487468593 + 3 = 0.512531407 t = 0.512531407 Simplifying t = 0.512531407Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.487468593, 0.512531407}
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