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Simplifying -16t2 + 80t + 145 = 0 Reorder the terms: 145 + 80t + -16t2 = 0 Solving 145 + 80t + -16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -9.0625 + -5t + t2 = 0 Move the constant term to the right: Add '9.0625' to each side of the equation. -9.0625 + -5t + 9.0625 + t2 = 0 + 9.0625 Reorder the terms: -9.0625 + 9.0625 + -5t + t2 = 0 + 9.0625 Combine like terms: -9.0625 + 9.0625 = 0.0000 0.0000 + -5t + t2 = 0 + 9.0625 -5t + t2 = 0 + 9.0625 Combine like terms: 0 + 9.0625 = 9.0625 -5t + t2 = 9.0625 The t term is -5t. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5t + 6.25 + t2 = 9.0625 + 6.25 Reorder the terms: 6.25 + -5t + t2 = 9.0625 + 6.25 Combine like terms: 9.0625 + 6.25 = 15.3125 6.25 + -5t + t2 = 15.3125 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 15.3125 Calculate the square root of the right side: 3.913118961 Break this problem into two subproblems by setting (t + -2.5) equal to 3.913118961 and -3.913118961.Subproblem 1
t + -2.5 = 3.913118961 Simplifying t + -2.5 = 3.913118961 Reorder the terms: -2.5 + t = 3.913118961 Solving -2.5 + t = 3.913118961 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = 3.913118961 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 3.913118961 + 2.5 t = 3.913118961 + 2.5 Combine like terms: 3.913118961 + 2.5 = 6.413118961 t = 6.413118961 Simplifying t = 6.413118961Subproblem 2
t + -2.5 = -3.913118961 Simplifying t + -2.5 = -3.913118961 Reorder the terms: -2.5 + t = -3.913118961 Solving -2.5 + t = -3.913118961 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + t = -3.913118961 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -3.913118961 + 2.5 t = -3.913118961 + 2.5 Combine like terms: -3.913118961 + 2.5 = -1.413118961 t = -1.413118961 Simplifying t = -1.413118961Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.413118961, -1.413118961}
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