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Simplifying 15 = -16t2 + 30t + 5 Reorder the terms: 15 = 5 + 30t + -16t2 Solving 15 = 5 + 30t + -16t2 Solving for variable 't'. Combine like terms: 15 + -5 = 10 10 + -30t + 16t2 = 5 + 30t + -16t2 + -5 + -30t + 16t2 Reorder the terms: 10 + -30t + 16t2 = 5 + -5 + 30t + -30t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 10 + -30t + 16t2 = 0 + 30t + -30t + -16t2 + 16t2 10 + -30t + 16t2 = 30t + -30t + -16t2 + 16t2 Combine like terms: 30t + -30t = 0 10 + -30t + 16t2 = 0 + -16t2 + 16t2 10 + -30t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 10 + -30t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(5 + -15t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(5 + -15t + 8t2)' equal to zero and attempt to solve: Simplifying 5 + -15t + 8t2 = 0 Solving 5 + -15t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.625 + -1.875t + t2 = 0 Move the constant term to the right: Add '-0.625' to each side of the equation. 0.625 + -1.875t + -0.625 + t2 = 0 + -0.625 Reorder the terms: 0.625 + -0.625 + -1.875t + t2 = 0 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + -1.875t + t2 = 0 + -0.625 -1.875t + t2 = 0 + -0.625 Combine like terms: 0 + -0.625 = -0.625 -1.875t + t2 = -0.625 The t term is -1.875t. Take half its coefficient (-0.9375). Square it (0.87890625) and add it to both sides. Add '0.87890625' to each side of the equation. -1.875t + 0.87890625 + t2 = -0.625 + 0.87890625 Reorder the terms: 0.87890625 + -1.875t + t2 = -0.625 + 0.87890625 Combine like terms: -0.625 + 0.87890625 = 0.25390625 0.87890625 + -1.875t + t2 = 0.25390625 Factor a perfect square on the left side: (t + -0.9375)(t + -0.9375) = 0.25390625 Calculate the square root of the right side: 0.503891109 Break this problem into two subproblems by setting (t + -0.9375) equal to 0.503891109 and -0.503891109.Subproblem 1
t + -0.9375 = 0.503891109 Simplifying t + -0.9375 = 0.503891109 Reorder the terms: -0.9375 + t = 0.503891109 Solving -0.9375 + t = 0.503891109 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.9375' to each side of the equation. -0.9375 + 0.9375 + t = 0.503891109 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = 0.503891109 + 0.9375 t = 0.503891109 + 0.9375 Combine like terms: 0.503891109 + 0.9375 = 1.441391109 t = 1.441391109 Simplifying t = 1.441391109Subproblem 2
t + -0.9375 = -0.503891109 Simplifying t + -0.9375 = -0.503891109 Reorder the terms: -0.9375 + t = -0.503891109 Solving -0.9375 + t = -0.503891109 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.9375' to each side of the equation. -0.9375 + 0.9375 + t = -0.503891109 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + t = -0.503891109 + 0.9375 t = -0.503891109 + 0.9375 Combine like terms: -0.503891109 + 0.9375 = 0.433608891 t = 0.433608891 Simplifying t = 0.433608891Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.441391109, 0.433608891}Solution
t = {1.441391109, 0.433608891}
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