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Simplifying -16t2 + 34t + 80 = 0 Reorder the terms: 80 + 34t + -16t2 = 0 Solving 80 + 34t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '2'. 2(40 + 17t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(40 + 17t + -8t2)' equal to zero and attempt to solve: Simplifying 40 + 17t + -8t2 = 0 Solving 40 + 17t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -5 + -2.125t + t2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -2.125t + 5 + t2 = 0 + 5 Reorder the terms: -5 + 5 + -2.125t + t2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -2.125t + t2 = 0 + 5 -2.125t + t2 = 0 + 5 Combine like terms: 0 + 5 = 5 -2.125t + t2 = 5 The t term is -2.125t. Take half its coefficient (-1.0625). Square it (1.12890625) and add it to both sides. Add '1.12890625' to each side of the equation. -2.125t + 1.12890625 + t2 = 5 + 1.12890625 Reorder the terms: 1.12890625 + -2.125t + t2 = 5 + 1.12890625 Combine like terms: 5 + 1.12890625 = 6.12890625 1.12890625 + -2.125t + t2 = 6.12890625 Factor a perfect square on the left side: (t + -1.0625)(t + -1.0625) = 6.12890625 Calculate the square root of the right side: 2.47566279 Break this problem into two subproblems by setting (t + -1.0625) equal to 2.47566279 and -2.47566279.Subproblem 1
t + -1.0625 = 2.47566279 Simplifying t + -1.0625 = 2.47566279 Reorder the terms: -1.0625 + t = 2.47566279 Solving -1.0625 + t = 2.47566279 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.0625' to each side of the equation. -1.0625 + 1.0625 + t = 2.47566279 + 1.0625 Combine like terms: -1.0625 + 1.0625 = 0.0000 0.0000 + t = 2.47566279 + 1.0625 t = 2.47566279 + 1.0625 Combine like terms: 2.47566279 + 1.0625 = 3.53816279 t = 3.53816279 Simplifying t = 3.53816279Subproblem 2
t + -1.0625 = -2.47566279 Simplifying t + -1.0625 = -2.47566279 Reorder the terms: -1.0625 + t = -2.47566279 Solving -1.0625 + t = -2.47566279 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.0625' to each side of the equation. -1.0625 + 1.0625 + t = -2.47566279 + 1.0625 Combine like terms: -1.0625 + 1.0625 = 0.0000 0.0000 + t = -2.47566279 + 1.0625 t = -2.47566279 + 1.0625 Combine like terms: -2.47566279 + 1.0625 = -1.41316279 t = -1.41316279 Simplifying t = -1.41316279Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.53816279, -1.41316279}Solution
t = {3.53816279, -1.41316279}
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