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Simplifying 16t2 + 32t = 1.9 Reorder the terms: 32t + 16t2 = 1.9 Solving 32t + 16t2 = 1.9 Solving for variable 't'. Reorder the terms: -1.9 + 32t + 16t2 = 1.9 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 -1.9 + 32t + 16t2 = 0.0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.11875 + 2t + t2 = 0.0 Move the constant term to the right: Add '0.11875' to each side of the equation. -0.11875 + 2t + 0.11875 + t2 = 0.0 + 0.11875 Reorder the terms: -0.11875 + 0.11875 + 2t + t2 = 0.0 + 0.11875 Combine like terms: -0.11875 + 0.11875 = 0.00000 0.00000 + 2t + t2 = 0.0 + 0.11875 2t + t2 = 0.0 + 0.11875 Combine like terms: 0.0 + 0.11875 = 0.11875 2t + t2 = 0.11875 The t term is 2t. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2t + 1 + t2 = 0.11875 + 1 Reorder the terms: 1 + 2t + t2 = 0.11875 + 1 Combine like terms: 0.11875 + 1 = 1.11875 1 + 2t + t2 = 1.11875 Factor a perfect square on the left side: (t + 1)(t + 1) = 1.11875 Calculate the square root of the right side: 1.05770979 Break this problem into two subproblems by setting (t + 1) equal to 1.05770979 and -1.05770979.Subproblem 1
t + 1 = 1.05770979 Simplifying t + 1 = 1.05770979 Reorder the terms: 1 + t = 1.05770979 Solving 1 + t = 1.05770979 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = 1.05770979 + -1 Combine like terms: 1 + -1 = 0 0 + t = 1.05770979 + -1 t = 1.05770979 + -1 Combine like terms: 1.05770979 + -1 = 0.05770979 t = 0.05770979 Simplifying t = 0.05770979Subproblem 2
t + 1 = -1.05770979 Simplifying t + 1 = -1.05770979 Reorder the terms: 1 + t = -1.05770979 Solving 1 + t = -1.05770979 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + t = -1.05770979 + -1 Combine like terms: 1 + -1 = 0 0 + t = -1.05770979 + -1 t = -1.05770979 + -1 Combine like terms: -1.05770979 + -1 = -2.05770979 t = -2.05770979 Simplifying t = -2.05770979Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.05770979, -2.05770979}
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