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Simplifying -16t2 + 30t + 2 = 10 Reorder the terms: 2 + 30t + -16t2 = 10 Solving 2 + 30t + -16t2 = 10 Solving for variable 't'. Reorder the terms: 2 + -10 + 30t + -16t2 = 10 + -10 Combine like terms: 2 + -10 = -8 -8 + 30t + -16t2 = 10 + -10 Combine like terms: 10 + -10 = 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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