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Simplifying 0 = -16t2 + 30t + 78 Reorder the terms: 0 = 78 + 30t + -16t2 Solving 0 = 78 + 30t + -16t2 Solving for variable 't'. Combine like terms: 0 + -78 = -78 -78 + -30t + 16t2 = 78 + 30t + -16t2 + -78 + -30t + 16t2 Reorder the terms: -78 + -30t + 16t2 = 78 + -78 + 30t + -30t + -16t2 + 16t2 Combine like terms: 78 + -78 = 0 -78 + -30t + 16t2 = 0 + 30t + -30t + -16t2 + 16t2 -78 + -30t + 16t2 = 30t + -30t + -16t2 + 16t2 Combine like terms: 30t + -30t = 0 -78 + -30t + 16t2 = 0 + -16t2 + 16t2 -78 + -30t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -78 + -30t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-39 + -15t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-39 + -15t + 8t2)' equal to zero and attempt to solve: Simplifying -39 + -15t + 8t2 = 0 Solving -39 + -15t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -4.875 + -1.875t + t2 = 0 Move the constant term to the right: Add '4.875' to each side of the equation. -4.875 + -1.875t + 4.875 + t2 = 0 + 4.875 Reorder the terms: -4.875 + 4.875 + -1.875t + t2 = 0 + 4.875 Combine like terms: -4.875 + 4.875 = 0.000 0.000 + -1.875t + t2 = 0 + 4.875 -1.875t + t2 = 0 + 4.875 Combine like terms: 0 + 4.875 = 4.875 -1.875t + t2 = 4.875 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 = 4.875 + 0.87890625 Reorder the terms: 0.87890625 + -1.875t + t2 = 4.875 + 0.87890625 Combine like terms: 4.875 + 0.87890625 = 5.75390625 0.87890625 + -1.875t + t2 = 5.75390625 Factor a perfect square on the left side: (t + -0.9375)(t + -0.9375) = 5.75390625 Calculate the square root of the right side: 2.398730133 Break this problem into two subproblems by setting (t + -0.9375) equal to 2.398730133 and -2.398730133.Subproblem 1
t + -0.9375 = 2.398730133 Simplifying t + -0.9375 = 2.398730133 Reorder the terms: -0.9375 + t = 2.398730133 Solving -0.9375 + t = 2.398730133 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 = 2.398730133 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = 2.398730133 + 0.9375 t = 2.398730133 + 0.9375 Combine like terms: 2.398730133 + 0.9375 = 3.336230133 t = 3.336230133 Simplifying t = 3.336230133Subproblem 2
t + -0.9375 = -2.398730133 Simplifying t + -0.9375 = -2.398730133 Reorder the terms: -0.9375 + t = -2.398730133 Solving -0.9375 + t = -2.398730133 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 = -2.398730133 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = -2.398730133 + 0.9375 t = -2.398730133 + 0.9375 Combine like terms: -2.398730133 + 0.9375 = -1.461230133 t = -1.461230133 Simplifying t = -1.461230133Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.336230133, -1.461230133}Solution
t = {3.336230133, -1.461230133}
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