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Simplifying -16t2 + -64t + 480 = 0 Reorder the terms: 480 + -64t + -16t2 = 0 Solving 480 + -64t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '16'. 16(30 + -4t + -1t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(30 + -4t + -1t2)' equal to zero and attempt to solve: Simplifying 30 + -4t + -1t2 = 0 Solving 30 + -4t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -30 + 4t + t2 = 0 Move the constant term to the right: Add '30' to each side of the equation. -30 + 4t + 30 + t2 = 0 + 30 Reorder the terms: -30 + 30 + 4t + t2 = 0 + 30 Combine like terms: -30 + 30 = 0 0 + 4t + t2 = 0 + 30 4t + t2 = 0 + 30 Combine like terms: 0 + 30 = 30 4t + t2 = 30 The t term is 4t. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4t + 4 + t2 = 30 + 4 Reorder the terms: 4 + 4t + t2 = 30 + 4 Combine like terms: 30 + 4 = 34 4 + 4t + t2 = 34 Factor a perfect square on the left side: (t + 2)(t + 2) = 34 Calculate the square root of the right side: 5.830951895 Break this problem into two subproblems by setting (t + 2) equal to 5.830951895 and -5.830951895.Subproblem 1
t + 2 = 5.830951895 Simplifying t + 2 = 5.830951895 Reorder the terms: 2 + t = 5.830951895 Solving 2 + t = 5.830951895 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + t = 5.830951895 + -2 Combine like terms: 2 + -2 = 0 0 + t = 5.830951895 + -2 t = 5.830951895 + -2 Combine like terms: 5.830951895 + -2 = 3.830951895 t = 3.830951895 Simplifying t = 3.830951895Subproblem 2
t + 2 = -5.830951895 Simplifying t + 2 = -5.830951895 Reorder the terms: 2 + t = -5.830951895 Solving 2 + t = -5.830951895 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + t = -5.830951895 + -2 Combine like terms: 2 + -2 = 0 0 + t = -5.830951895 + -2 t = -5.830951895 + -2 Combine like terms: -5.830951895 + -2 = -7.830951895 t = -7.830951895 Simplifying t = -7.830951895Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.830951895, -7.830951895}Solution
t = {3.830951895, -7.830951895}
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