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Simplifying 0 = 4 + 40t + -16t2 Solving 0 = 4 + 40t + -16t2 Solving for variable 't'. Combine like terms: 0 + -4 = -4 -4 + -40t + 16t2 = 4 + 40t + -16t2 + -4 + -40t + 16t2 Reorder the terms: -4 + -40t + 16t2 = 4 + -4 + 40t + -40t + -16t2 + 16t2 Combine like terms: 4 + -4 = 0 -4 + -40t + 16t2 = 0 + 40t + -40t + -16t2 + 16t2 -4 + -40t + 16t2 = 40t + -40t + -16t2 + 16t2 Combine like terms: 40t + -40t = 0 -4 + -40t + 16t2 = 0 + -16t2 + 16t2 -4 + -40t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -4 + -40t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-1 + -10t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-1 + -10t + 4t2)' equal to zero and attempt to solve: Simplifying -1 + -10t + 4t2 = 0 Solving -1 + -10t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + -2.5t + t2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -2.5t + 0.25 + t2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -2.5t + t2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -2.5t + t2 = 0 + 0.25 -2.5t + t2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -2.5t + t2 = 0.25 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 = 0.25 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = 0.25 + 1.5625 Combine like terms: 0.25 + 1.5625 = 1.8125 1.5625 + -2.5t + t2 = 1.8125 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 1.8125 Calculate the square root of the right side: 1.346291202 Break this problem into two subproblems by setting (t + -1.25) equal to 1.346291202 and -1.346291202.Subproblem 1
t + -1.25 = 1.346291202 Simplifying t + -1.25 = 1.346291202 Reorder the terms: -1.25 + t = 1.346291202 Solving -1.25 + t = 1.346291202 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 = 1.346291202 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 1.346291202 + 1.25 t = 1.346291202 + 1.25 Combine like terms: 1.346291202 + 1.25 = 2.596291202 t = 2.596291202 Simplifying t = 2.596291202Subproblem 2
t + -1.25 = -1.346291202 Simplifying t + -1.25 = -1.346291202 Reorder the terms: -1.25 + t = -1.346291202 Solving -1.25 + t = -1.346291202 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 = -1.346291202 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -1.346291202 + 1.25 t = -1.346291202 + 1.25 Combine like terms: -1.346291202 + 1.25 = -0.096291202 t = -0.096291202 Simplifying t = -0.096291202Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.596291202, -0.096291202}Solution
t = {2.596291202, -0.096291202}
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