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Simplifying 0 = 16t2 + 48t + 10 Reorder the terms: 0 = 10 + 48t + 16t2 Solving 0 = 10 + 48t + 16t2 Solving for variable 't'. Combine like terms: 0 + -10 = -10 -10 + -48t + -16t2 = 10 + 48t + 16t2 + -10 + -48t + -16t2 Reorder the terms: -10 + -48t + -16t2 = 10 + -10 + 48t + -48t + 16t2 + -16t2 Combine like terms: 10 + -10 = 0 -10 + -48t + -16t2 = 0 + 48t + -48t + 16t2 + -16t2 -10 + -48t + -16t2 = 48t + -48t + 16t2 + -16t2 Combine like terms: 48t + -48t = 0 -10 + -48t + -16t2 = 0 + 16t2 + -16t2 -10 + -48t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -10 + -48t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-2'. -2(5 + 24t + 8t2) = 0 Ignore the factor -2.Subproblem 1
Set the factor '(5 + 24t + 8t2)' equal to zero and attempt to solve: Simplifying 5 + 24t + 8t2 = 0 Solving 5 + 24t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.625 + 3t + t2 = 0 Move the constant term to the right: Add '-0.625' to each side of the equation. 0.625 + 3t + -0.625 + t2 = 0 + -0.625 Reorder the terms: 0.625 + -0.625 + 3t + t2 = 0 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + 3t + t2 = 0 + -0.625 3t + t2 = 0 + -0.625 Combine like terms: 0 + -0.625 = -0.625 3t + t2 = -0.625 The t term is 3t. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3t + 2.25 + t2 = -0.625 + 2.25 Reorder the terms: 2.25 + 3t + t2 = -0.625 + 2.25 Combine like terms: -0.625 + 2.25 = 1.625 2.25 + 3t + t2 = 1.625 Factor a perfect square on the left side: (t + 1.5)(t + 1.5) = 1.625 Calculate the square root of the right side: 1.274754878 Break this problem into two subproblems by setting (t + 1.5) equal to 1.274754878 and -1.274754878.Subproblem 1
t + 1.5 = 1.274754878 Simplifying t + 1.5 = 1.274754878 Reorder the terms: 1.5 + t = 1.274754878 Solving 1.5 + t = 1.274754878 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = 1.274754878 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = 1.274754878 + -1.5 t = 1.274754878 + -1.5 Combine like terms: 1.274754878 + -1.5 = -0.225245122 t = -0.225245122 Simplifying t = -0.225245122Subproblem 2
t + 1.5 = -1.274754878 Simplifying t + 1.5 = -1.274754878 Reorder the terms: 1.5 + t = -1.274754878 Solving 1.5 + t = -1.274754878 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = -1.274754878 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = -1.274754878 + -1.5 t = -1.274754878 + -1.5 Combine like terms: -1.274754878 + -1.5 = -2.774754878 t = -2.774754878 Simplifying t = -2.774754878Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.225245122, -2.774754878}Solution
t = {-0.225245122, -2.774754878}
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