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Simplifying -16t2 + 40t + 80 = 0 Reorder the terms: 80 + 40t + -16t2 = 0 Solving 80 + 40t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '8'. 8(10 + 5t + -2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(10 + 5t + -2t2)' equal to zero and attempt to solve: Simplifying 10 + 5t + -2t2 = 0 Solving 10 + 5t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -5 + -2.5t + t2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -2.5t + 5 + t2 = 0 + 5 Reorder the terms: -5 + 5 + -2.5t + t2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -2.5t + t2 = 0 + 5 -2.5t + t2 = 0 + 5 Combine like terms: 0 + 5 = 5 -2.5t + t2 = 5 The t term is -2.5t. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5t + 1.5625 + t2 = 5 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = 5 + 1.5625 Combine like terms: 5 + 1.5625 = 6.5625 1.5625 + -2.5t + t2 = 6.5625 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 6.5625 Calculate the square root of the right side: 2.561737691 Break this problem into two subproblems by setting (t + -1.25) equal to 2.561737691 and -2.561737691.Subproblem 1
t + -1.25 = 2.561737691 Simplifying t + -1.25 = 2.561737691 Reorder the terms: -1.25 + t = 2.561737691 Solving -1.25 + t = 2.561737691 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = 2.561737691 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 2.561737691 + 1.25 t = 2.561737691 + 1.25 Combine like terms: 2.561737691 + 1.25 = 3.811737691 t = 3.811737691 Simplifying t = 3.811737691Subproblem 2
t + -1.25 = -2.561737691 Simplifying t + -1.25 = -2.561737691 Reorder the terms: -1.25 + t = -2.561737691 Solving -1.25 + t = -2.561737691 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = -2.561737691 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -2.561737691 + 1.25 t = -2.561737691 + 1.25 Combine like terms: -2.561737691 + 1.25 = -1.311737691 t = -1.311737691 Simplifying t = -1.311737691Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.811737691, -1.311737691}Solution
t = {3.811737691, -1.311737691}
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