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Simplifying 424 = -16t2 + -6t + 480 Reorder the terms: 424 = 480 + -6t + -16t2 Solving 424 = 480 + -6t + -16t2 Solving for variable 't'. Combine like terms: 424 + -480 = -56 -56 + 6t + 16t2 = 480 + -6t + -16t2 + -480 + 6t + 16t2 Reorder the terms: -56 + 6t + 16t2 = 480 + -480 + -6t + 6t + -16t2 + 16t2 Combine like terms: 480 + -480 = 0 -56 + 6t + 16t2 = 0 + -6t + 6t + -16t2 + 16t2 -56 + 6t + 16t2 = -6t + 6t + -16t2 + 16t2 Combine like terms: -6t + 6t = 0 -56 + 6t + 16t2 = 0 + -16t2 + 16t2 -56 + 6t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -56 + 6t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-28 + 3t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-28 + 3t + 8t2)' equal to zero and attempt to solve: Simplifying -28 + 3t + 8t2 = 0 Solving -28 + 3t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -3.5 + 0.375t + t2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + 0.375t + 3.5 + t2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + 0.375t + t2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + 0.375t + t2 = 0 + 3.5 0.375t + t2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 0.375t + t2 = 3.5 The t term is 0.375t. Take half its coefficient (0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. 0.375t + 0.03515625 + t2 = 3.5 + 0.03515625 Reorder the terms: 0.03515625 + 0.375t + t2 = 3.5 + 0.03515625 Combine like terms: 3.5 + 0.03515625 = 3.53515625 0.03515625 + 0.375t + t2 = 3.53515625 Factor a perfect square on the left side: (t + 0.1875)(t + 0.1875) = 3.53515625 Calculate the square root of the right side: 1.88020112 Break this problem into two subproblems by setting (t + 0.1875) equal to 1.88020112 and -1.88020112.Subproblem 1
t + 0.1875 = 1.88020112 Simplifying t + 0.1875 = 1.88020112 Reorder the terms: 0.1875 + t = 1.88020112 Solving 0.1875 + t = 1.88020112 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + t = 1.88020112 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + t = 1.88020112 + -0.1875 t = 1.88020112 + -0.1875 Combine like terms: 1.88020112 + -0.1875 = 1.69270112 t = 1.69270112 Simplifying t = 1.69270112Subproblem 2
t + 0.1875 = -1.88020112 Simplifying t + 0.1875 = -1.88020112 Reorder the terms: 0.1875 + t = -1.88020112 Solving 0.1875 + t = -1.88020112 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + t = -1.88020112 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + t = -1.88020112 + -0.1875 t = -1.88020112 + -0.1875 Combine like terms: -1.88020112 + -0.1875 = -2.06770112 t = -2.06770112 Simplifying t = -2.06770112Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.69270112, -2.06770112}Solution
t = {1.69270112, -2.06770112}
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