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Simplifying 0 = -16t2 + 60t + 40 Reorder the terms: 0 = 40 + 60t + -16t2 Solving 0 = 40 + 60t + -16t2 Solving for variable 't'. Combine like terms: 0 + -40 = -40 -40 + -60t + 16t2 = 40 + 60t + -16t2 + -40 + -60t + 16t2 Reorder the terms: -40 + -60t + 16t2 = 40 + -40 + 60t + -60t + -16t2 + 16t2 Combine like terms: 40 + -40 = 0 -40 + -60t + 16t2 = 0 + 60t + -60t + -16t2 + 16t2 -40 + -60t + 16t2 = 60t + -60t + -16t2 + 16t2 Combine like terms: 60t + -60t = 0 -40 + -60t + 16t2 = 0 + -16t2 + 16t2 -40 + -60t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -40 + -60t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-10 + -15t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-10 + -15t + 4t2)' equal to zero and attempt to solve: Simplifying -10 + -15t + 4t2 = 0 Solving -10 + -15t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2.5 + -3.75t + t2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -3.75t + 2.5 + t2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -3.75t + t2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -3.75t + t2 = 0 + 2.5 -3.75t + t2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -3.75t + t2 = 2.5 The t term is -3.75t. Take half its coefficient (-1.875). Square it (3.515625) and add it to both sides. Add '3.515625' to each side of the equation. -3.75t + 3.515625 + t2 = 2.5 + 3.515625 Reorder the terms: 3.515625 + -3.75t + t2 = 2.5 + 3.515625 Combine like terms: 2.5 + 3.515625 = 6.015625 3.515625 + -3.75t + t2 = 6.015625 Factor a perfect square on the left side: (t + -1.875)(t + -1.875) = 6.015625 Calculate the square root of the right side: 2.452677109 Break this problem into two subproblems by setting (t + -1.875) equal to 2.452677109 and -2.452677109.Subproblem 1
t + -1.875 = 2.452677109 Simplifying t + -1.875 = 2.452677109 Reorder the terms: -1.875 + t = 2.452677109 Solving -1.875 + t = 2.452677109 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + t = 2.452677109 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + t = 2.452677109 + 1.875 t = 2.452677109 + 1.875 Combine like terms: 2.452677109 + 1.875 = 4.327677109 t = 4.327677109 Simplifying t = 4.327677109Subproblem 2
t + -1.875 = -2.452677109 Simplifying t + -1.875 = -2.452677109 Reorder the terms: -1.875 + t = -2.452677109 Solving -1.875 + t = -2.452677109 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + t = -2.452677109 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + t = -2.452677109 + 1.875 t = -2.452677109 + 1.875 Combine like terms: -2.452677109 + 1.875 = -0.577677109 t = -0.577677109 Simplifying t = -0.577677109Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.327677109, -0.577677109}Solution
t = {4.327677109, -0.577677109}
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