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Simplifying -16t2 + -40t + -20 = 0 Reorder the terms: -20 + -40t + -16t2 = 0 Solving -20 + -40t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '-4'. -4(5 + 10t + 4t2) = 0 Ignore the factor -4.Subproblem 1
Set the factor '(5 + 10t + 4t2)' equal to zero and attempt to solve: Simplifying 5 + 10t + 4t2 = 0 Solving 5 + 10t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1.25 + 2.5t + t2 = 0 Move the constant term to the right: Add '-1.25' to each side of the equation. 1.25 + 2.5t + -1.25 + t2 = 0 + -1.25 Reorder the terms: 1.25 + -1.25 + 2.5t + t2 = 0 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + 2.5t + t2 = 0 + -1.25 2.5t + t2 = 0 + -1.25 Combine like terms: 0 + -1.25 = -1.25 2.5t + t2 = -1.25 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 = -1.25 + 1.5625 Reorder the terms: 1.5625 + 2.5t + t2 = -1.25 + 1.5625 Combine like terms: -1.25 + 1.5625 = 0.3125 1.5625 + 2.5t + t2 = 0.3125 Factor a perfect square on the left side: (t + 1.25)(t + 1.25) = 0.3125 Calculate the square root of the right side: 0.559016994 Break this problem into two subproblems by setting (t + 1.25) equal to 0.559016994 and -0.559016994.Subproblem 1
t + 1.25 = 0.559016994 Simplifying t + 1.25 = 0.559016994 Reorder the terms: 1.25 + t = 0.559016994 Solving 1.25 + t = 0.559016994 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 = 0.559016994 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + t = 0.559016994 + -1.25 t = 0.559016994 + -1.25 Combine like terms: 0.559016994 + -1.25 = -0.690983006 t = -0.690983006 Simplifying t = -0.690983006Subproblem 2
t + 1.25 = -0.559016994 Simplifying t + 1.25 = -0.559016994 Reorder the terms: 1.25 + t = -0.559016994 Solving 1.25 + t = -0.559016994 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 = -0.559016994 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + t = -0.559016994 + -1.25 t = -0.559016994 + -1.25 Combine like terms: -0.559016994 + -1.25 = -1.809016994 t = -1.809016994 Simplifying t = -1.809016994Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.690983006, -1.809016994}Solution
t = {-0.690983006, -1.809016994}
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