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Simplifying 446 = -16t2 + -4t + 500 Reorder the terms: 446 = 500 + -4t + -16t2 Solving 446 = 500 + -4t + -16t2 Solving for variable 't'. Combine like terms: 446 + -500 = -54 -54 + 4t + 16t2 = 500 + -4t + -16t2 + -500 + 4t + 16t2 Reorder the terms: -54 + 4t + 16t2 = 500 + -500 + -4t + 4t + -16t2 + 16t2 Combine like terms: 500 + -500 = 0 -54 + 4t + 16t2 = 0 + -4t + 4t + -16t2 + 16t2 -54 + 4t + 16t2 = -4t + 4t + -16t2 + 16t2 Combine like terms: -4t + 4t = 0 -54 + 4t + 16t2 = 0 + -16t2 + 16t2 -54 + 4t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -54 + 4t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-27 + 2t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-27 + 2t + 8t2)' equal to zero and attempt to solve: Simplifying -27 + 2t + 8t2 = 0 Solving -27 + 2t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -3.375 + 0.25t + t2 = 0 Move the constant term to the right: Add '3.375' to each side of the equation. -3.375 + 0.25t + 3.375 + t2 = 0 + 3.375 Reorder the terms: -3.375 + 3.375 + 0.25t + t2 = 0 + 3.375 Combine like terms: -3.375 + 3.375 = 0.000 0.000 + 0.25t + t2 = 0 + 3.375 0.25t + t2 = 0 + 3.375 Combine like terms: 0 + 3.375 = 3.375 0.25t + t2 = 3.375 The t term is 0.25t. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25t + 0.015625 + t2 = 3.375 + 0.015625 Reorder the terms: 0.015625 + 0.25t + t2 = 3.375 + 0.015625 Combine like terms: 3.375 + 0.015625 = 3.390625 0.015625 + 0.25t + t2 = 3.390625 Factor a perfect square on the left side: (t + 0.125)(t + 0.125) = 3.390625 Calculate the square root of the right side: 1.841364983 Break this problem into two subproblems by setting (t + 0.125) equal to 1.841364983 and -1.841364983.Subproblem 1
t + 0.125 = 1.841364983 Simplifying t + 0.125 = 1.841364983 Reorder the terms: 0.125 + t = 1.841364983 Solving 0.125 + t = 1.841364983 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + t = 1.841364983 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + t = 1.841364983 + -0.125 t = 1.841364983 + -0.125 Combine like terms: 1.841364983 + -0.125 = 1.716364983 t = 1.716364983 Simplifying t = 1.716364983Subproblem 2
t + 0.125 = -1.841364983 Simplifying t + 0.125 = -1.841364983 Reorder the terms: 0.125 + t = -1.841364983 Solving 0.125 + t = -1.841364983 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + t = -1.841364983 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + t = -1.841364983 + -0.125 t = -1.841364983 + -0.125 Combine like terms: -1.841364983 + -0.125 = -1.966364983 t = -1.966364983 Simplifying t = -1.966364983Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.716364983, -1.966364983}Solution
t = {1.716364983, -1.966364983}
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