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Simplifying 40t = -16t2 + 24t + 40 Reorder the terms: 40t = 40 + 24t + -16t2 Solving 40t = 40 + 24t + -16t2 Solving for variable 't'. Reorder the terms: -40 + 40t + -24t + 16t2 = 40 + 24t + -16t2 + -40 + -24t + 16t2 Combine like terms: 40t + -24t = 16t -40 + 16t + 16t2 = 40 + 24t + -16t2 + -40 + -24t + 16t2 Reorder the terms: -40 + 16t + 16t2 = 40 + -40 + 24t + -24t + -16t2 + 16t2 Combine like terms: 40 + -40 = 0 -40 + 16t + 16t2 = 0 + 24t + -24t + -16t2 + 16t2 -40 + 16t + 16t2 = 24t + -24t + -16t2 + 16t2 Combine like terms: 24t + -24t = 0 -40 + 16t + 16t2 = 0 + -16t2 + 16t2 -40 + 16t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -40 + 16t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-5 + 2t + 2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-5 + 2t + 2t2)' equal to zero and attempt to solve: Simplifying -5 + 2t + 2t2 = 0 Solving -5 + 2t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2.5 + t + t2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + t + 2.5 + t2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + t + t2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t + t2 = 0 + 2.5 t + t2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 t + t2 = 2.5 The t term is t. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + t2 = 2.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + t2 = 2.5 + 0.25 Combine like terms: 2.5 + 0.25 = 2.75 1.25 + t2 = 2.75 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 2.75 Calculate the square root of the right side: 1.658312395 Break this problem into two subproblems by setting (t + 0.5) equal to 1.658312395 and -1.658312395.Subproblem 1
t + 0.5 = 1.658312395 Simplifying t + 0.5 = 1.658312395 Reorder the terms: 0.5 + t = 1.658312395 Solving 0.5 + t = 1.658312395 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 1.658312395 + -0.5 t = 1.658312395 + -0.5 Combine like terms: 1.658312395 + -0.5 = 1.158312395 t = 1.158312395 Simplifying t = 1.158312395Subproblem 2
t + 0.5 = -1.658312395 Simplifying t + 0.5 = -1.658312395 Reorder the terms: 0.5 + t = -1.658312395 Solving 0.5 + t = -1.658312395 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -1.658312395 + -0.5 t = -1.658312395 + -0.5 Combine like terms: -1.658312395 + -0.5 = -2.158312395 t = -2.158312395 Simplifying t = -2.158312395Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.158312395, -2.158312395}Solution
t = {1.158312395, -2.158312395}
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