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Simplifying 2.5 = -16t2 + 80t + 5 Reorder the terms: 2.5 = 5 + 80t + -16t2 Solving 2.5 = 5 + 80t + -16t2 Solving for variable 't'. Combine like terms: 2.5 + -5 = -2.5 -2.5 + -80t + 16t2 = 5 + 80t + -16t2 + -5 + -80t + 16t2 Reorder the terms: -2.5 + -80t + 16t2 = 5 + -5 + 80t + -80t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 -2.5 + -80t + 16t2 = 0 + 80t + -80t + -16t2 + 16t2 -2.5 + -80t + 16t2 = 80t + -80t + -16t2 + 16t2 Combine like terms: 80t + -80t = 0 -2.5 + -80t + 16t2 = 0 + -16t2 + 16t2 -2.5 + -80t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -2.5 + -80t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.15625 + -5t + t2 = 0 Move the constant term to the right: Add '0.15625' to each side of the equation. -0.15625 + -5t + 0.15625 + t2 = 0 + 0.15625 Reorder the terms: -0.15625 + 0.15625 + -5t + t2 = 0 + 0.15625 Combine like terms: -0.15625 + 0.15625 = 0.00000 0.00000 + -5t + t2 = 0 + 0.15625 -5t + t2 = 0 + 0.15625 Combine like terms: 0 + 0.15625 = 0.15625 -5t + t2 = 0.15625 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 = 0.15625 + 6.25 Reorder the terms: 6.25 + -5t + t2 = 0.15625 + 6.25 Combine like terms: 0.15625 + 6.25 = 6.40625 6.25 + -5t + t2 = 6.40625 Factor a perfect square on the left side: (t + -2.5)(t + -2.5) = 6.40625 Calculate the square root of the right side: 2.531057091 Break this problem into two subproblems by setting (t + -2.5) equal to 2.531057091 and -2.531057091.Subproblem 1
t + -2.5 = 2.531057091 Simplifying t + -2.5 = 2.531057091 Reorder the terms: -2.5 + t = 2.531057091 Solving -2.5 + t = 2.531057091 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 = 2.531057091 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = 2.531057091 + 2.5 t = 2.531057091 + 2.5 Combine like terms: 2.531057091 + 2.5 = 5.031057091 t = 5.031057091 Simplifying t = 5.031057091Subproblem 2
t + -2.5 = -2.531057091 Simplifying t + -2.5 = -2.531057091 Reorder the terms: -2.5 + t = -2.531057091 Solving -2.5 + t = -2.531057091 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 = -2.531057091 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + t = -2.531057091 + 2.5 t = -2.531057091 + 2.5 Combine like terms: -2.531057091 + 2.5 = -0.031057091 t = -0.031057091 Simplifying t = -0.031057091Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.031057091, -0.031057091}
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