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Simplifying 40(t) = -16t2 + 25(t) + 6 Reorder the terms: 40t = 6 + 25t + -16t2 Solving 40t = 6 + 25t + -16t2 Solving for variable 't'. Reorder the terms: -6 + 40t + -25t + 16t2 = 6 + 25t + -16t2 + -6 + -25t + 16t2 Combine like terms: 40t + -25t = 15t -6 + 15t + 16t2 = 6 + 25t + -16t2 + -6 + -25t + 16t2 Reorder the terms: -6 + 15t + 16t2 = 6 + -6 + 25t + -25t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 -6 + 15t + 16t2 = 0 + 25t + -25t + -16t2 + 16t2 -6 + 15t + 16t2 = 25t + -25t + -16t2 + 16t2 Combine like terms: 25t + -25t = 0 -6 + 15t + 16t2 = 0 + -16t2 + 16t2 -6 + 15t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -6 + 15t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.375 + 0.9375t + t2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + 0.9375t + 0.375 + t2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + 0.9375t + t2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + 0.9375t + t2 = 0 + 0.375 0.9375t + t2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 0.9375t + t2 = 0.375 The t term is 0.9375t. Take half its coefficient (0.46875). Square it (0.2197265625) and add it to both sides. Add '0.2197265625' to each side of the equation. 0.9375t + 0.2197265625 + t2 = 0.375 + 0.2197265625 Reorder the terms: 0.2197265625 + 0.9375t + t2 = 0.375 + 0.2197265625 Combine like terms: 0.375 + 0.2197265625 = 0.5947265625 0.2197265625 + 0.9375t + t2 = 0.5947265625 Factor a perfect square on the left side: (t + 0.46875)(t + 0.46875) = 0.5947265625 Calculate the square root of the right side: 0.771185167 Break this problem into two subproblems by setting (t + 0.46875) equal to 0.771185167 and -0.771185167.Subproblem 1
t + 0.46875 = 0.771185167 Simplifying t + 0.46875 = 0.771185167 Reorder the terms: 0.46875 + t = 0.771185167 Solving 0.46875 + t = 0.771185167 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.46875' to each side of the equation. 0.46875 + -0.46875 + t = 0.771185167 + -0.46875 Combine like terms: 0.46875 + -0.46875 = 0.00000 0.00000 + t = 0.771185167 + -0.46875 t = 0.771185167 + -0.46875 Combine like terms: 0.771185167 + -0.46875 = 0.302435167 t = 0.302435167 Simplifying t = 0.302435167Subproblem 2
t + 0.46875 = -0.771185167 Simplifying t + 0.46875 = -0.771185167 Reorder the terms: 0.46875 + t = -0.771185167 Solving 0.46875 + t = -0.771185167 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.46875' to each side of the equation. 0.46875 + -0.46875 + t = -0.771185167 + -0.46875 Combine like terms: 0.46875 + -0.46875 = 0.00000 0.00000 + t = -0.771185167 + -0.46875 t = -0.771185167 + -0.46875 Combine like terms: -0.771185167 + -0.46875 = -1.239935167 t = -1.239935167 Simplifying t = -1.239935167Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.302435167, -1.239935167}
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