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Simplifying 15 = -16t2 + 40t + 3 Reorder the terms: 15 = 3 + 40t + -16t2 Solving 15 = 3 + 40t + -16t2 Solving for variable 't'. Combine like terms: 15 + -3 = 12 12 + -40t + 16t2 = 3 + 40t + -16t2 + -3 + -40t + 16t2 Reorder the terms: 12 + -40t + 16t2 = 3 + -3 + 40t + -40t + -16t2 + 16t2 Combine like terms: 3 + -3 = 0 12 + -40t + 16t2 = 0 + 40t + -40t + -16t2 + 16t2 12 + -40t + 16t2 = 40t + -40t + -16t2 + 16t2 Combine like terms: 40t + -40t = 0 12 + -40t + 16t2 = 0 + -16t2 + 16t2 12 + -40t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 12 + -40t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(3 + -10t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(3 + -10t + 4t2)' equal to zero and attempt to solve: Simplifying 3 + -10t + 4t2 = 0 Solving 3 + -10t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.75 + -2.5t + t2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + -2.5t + -0.75 + t2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + -2.5t + t2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + -2.5t + t2 = 0 + -0.75 -2.5t + t2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 -2.5t + t2 = -0.75 The t term is -2.5t. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5t + 1.5625 + t2 = -0.75 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = -0.75 + 1.5625 Combine like terms: -0.75 + 1.5625 = 0.8125 1.5625 + -2.5t + t2 = 0.8125 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 0.8125 Calculate the square root of the right side: 0.901387819 Break this problem into two subproblems by setting (t + -1.25) equal to 0.901387819 and -0.901387819.Subproblem 1
t + -1.25 = 0.901387819 Simplifying t + -1.25 = 0.901387819 Reorder the terms: -1.25 + t = 0.901387819 Solving -1.25 + t = 0.901387819 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = 0.901387819 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 0.901387819 + 1.25 t = 0.901387819 + 1.25 Combine like terms: 0.901387819 + 1.25 = 2.151387819 t = 2.151387819 Simplifying t = 2.151387819Subproblem 2
t + -1.25 = -0.901387819 Simplifying t + -1.25 = -0.901387819 Reorder the terms: -1.25 + t = -0.901387819 Solving -1.25 + t = -0.901387819 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + t = -0.901387819 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -0.901387819 + 1.25 t = -0.901387819 + 1.25 Combine like terms: -0.901387819 + 1.25 = 0.348612181 t = 0.348612181 Simplifying t = 0.348612181Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.151387819, 0.348612181}Solution
t = {2.151387819, 0.348612181}
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