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Simplifying 19 = -6t2 + 24t + 14 Reorder the terms: 19 = 14 + 24t + -6t2 Solving 19 = 14 + 24t + -6t2 Solving for variable 't'. Combine like terms: 19 + -14 = 5 5 + -24t + 6t2 = 14 + 24t + -6t2 + -14 + -24t + 6t2 Reorder the terms: 5 + -24t + 6t2 = 14 + -14 + 24t + -24t + -6t2 + 6t2 Combine like terms: 14 + -14 = 0 5 + -24t + 6t2 = 0 + 24t + -24t + -6t2 + 6t2 5 + -24t + 6t2 = 24t + -24t + -6t2 + 6t2 Combine like terms: 24t + -24t = 0 5 + -24t + 6t2 = 0 + -6t2 + 6t2 5 + -24t + 6t2 = -6t2 + 6t2 Combine like terms: -6t2 + 6t2 = 0 5 + -24t + 6t2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.8333333333 + -4t + t2 = 0 Move the constant term to the right: Add '-0.8333333333' to each side of the equation. 0.8333333333 + -4t + -0.8333333333 + t2 = 0 + -0.8333333333 Reorder the terms: 0.8333333333 + -0.8333333333 + -4t + t2 = 0 + -0.8333333333 Combine like terms: 0.8333333333 + -0.8333333333 = 0.0000000000 0.0000000000 + -4t + t2 = 0 + -0.8333333333 -4t + t2 = 0 + -0.8333333333 Combine like terms: 0 + -0.8333333333 = -0.8333333333 -4t + t2 = -0.8333333333 The t term is -4t. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4t + 4 + t2 = -0.8333333333 + 4 Reorder the terms: 4 + -4t + t2 = -0.8333333333 + 4 Combine like terms: -0.8333333333 + 4 = 3.1666666667 4 + -4t + t2 = 3.1666666667 Factor a perfect square on the left side: (t + -2)(t + -2) = 3.1666666667 Calculate the square root of the right side: 1.779513042 Break this problem into two subproblems by setting (t + -2) equal to 1.779513042 and -1.779513042.Subproblem 1
t + -2 = 1.779513042 Simplifying t + -2 = 1.779513042 Reorder the terms: -2 + t = 1.779513042 Solving -2 + t = 1.779513042 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = 1.779513042 + 2 Combine like terms: -2 + 2 = 0 0 + t = 1.779513042 + 2 t = 1.779513042 + 2 Combine like terms: 1.779513042 + 2 = 3.779513042 t = 3.779513042 Simplifying t = 3.779513042Subproblem 2
t + -2 = -1.779513042 Simplifying t + -2 = -1.779513042 Reorder the terms: -2 + t = -1.779513042 Solving -2 + t = -1.779513042 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + t = -1.779513042 + 2 Combine like terms: -2 + 2 = 0 0 + t = -1.779513042 + 2 t = -1.779513042 + 2 Combine like terms: -1.779513042 + 2 = 0.220486958 t = 0.220486958 Simplifying t = 0.220486958Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.779513042, 0.220486958}
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