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Simplifying 14 = 6 + 30t + -16t2 Solving 14 = 6 + 30t + -16t2 Solving for variable 't'. Combine like terms: 14 + -6 = 8 8 + -30t + 16t2 = 6 + 30t + -16t2 + -6 + -30t + 16t2 Reorder the terms: 8 + -30t + 16t2 = 6 + -6 + 30t + -30t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 8 + -30t + 16t2 = 0 + 30t + -30t + -16t2 + 16t2 8 + -30t + 16t2 = 30t + -30t + -16t2 + 16t2 Combine like terms: 30t + -30t = 0 8 + -30t + 16t2 = 0 + -16t2 + 16t2 8 + -30t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 8 + -30t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(4 + -15t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(4 + -15t + 8t2)' equal to zero and attempt to solve: Simplifying 4 + -15t + 8t2 = 0 Solving 4 + -15t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.5 + -1.875t + t2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -1.875t + -0.5 + t2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -1.875t + t2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -1.875t + t2 = 0 + -0.5 -1.875t + t2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -1.875t + t2 = -0.5 The t term is -1.875t. Take half its coefficient (-0.9375). Square it (0.87890625) and add it to both sides. Add '0.87890625' to each side of the equation. -1.875t + 0.87890625 + t2 = -0.5 + 0.87890625 Reorder the terms: 0.87890625 + -1.875t + t2 = -0.5 + 0.87890625 Combine like terms: -0.5 + 0.87890625 = 0.37890625 0.87890625 + -1.875t + t2 = 0.37890625 Factor a perfect square on the left side: (t + -0.9375)(t + -0.9375) = 0.37890625 Calculate the square root of the right side: 0.615553613 Break this problem into two subproblems by setting (t + -0.9375) equal to 0.615553613 and -0.615553613.Subproblem 1
t + -0.9375 = 0.615553613 Simplifying t + -0.9375 = 0.615553613 Reorder the terms: -0.9375 + t = 0.615553613 Solving -0.9375 + t = 0.615553613 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.9375' to each side of the equation. -0.9375 + 0.9375 + t = 0.615553613 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = 0.615553613 + 0.9375 t = 0.615553613 + 0.9375 Combine like terms: 0.615553613 + 0.9375 = 1.553053613 t = 1.553053613 Simplifying t = 1.553053613Subproblem 2
t + -0.9375 = -0.615553613 Simplifying t + -0.9375 = -0.615553613 Reorder the terms: -0.9375 + t = -0.615553613 Solving -0.9375 + t = -0.615553613 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.9375' to each side of the equation. -0.9375 + 0.9375 + t = -0.615553613 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = -0.615553613 + 0.9375 t = -0.615553613 + 0.9375 Combine like terms: -0.615553613 + 0.9375 = 0.321946387 t = 0.321946387 Simplifying t = 0.321946387Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.553053613, 0.321946387}Solution
t = {1.553053613, 0.321946387}
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