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Simplifying 8 = -16t2 + 40t + 40 Reorder the terms: 8 = 40 + 40t + -16t2 Solving 8 = 40 + 40t + -16t2 Solving for variable 't'. Combine like terms: 8 + -40 = -32 -32 + -40t + 16t2 = 40 + 40t + -16t2 + -40 + -40t + 16t2 Reorder the terms: -32 + -40t + 16t2 = 40 + -40 + 40t + -40t + -16t2 + 16t2 Combine like terms: 40 + -40 = 0 -32 + -40t + 16t2 = 0 + 40t + -40t + -16t2 + 16t2 -32 + -40t + 16t2 = 40t + -40t + -16t2 + 16t2 Combine like terms: 40t + -40t = 0 -32 + -40t + 16t2 = 0 + -16t2 + 16t2 -32 + -40t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -32 + -40t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-4 + -5t + 2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-4 + -5t + 2t2)' equal to zero and attempt to solve: Simplifying -4 + -5t + 2t2 = 0 Solving -4 + -5t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2 + -2.5t + t2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -2.5t + 2 + t2 = 0 + 2 Reorder the terms: -2 + 2 + -2.5t + t2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -2.5t + t2 = 0 + 2 -2.5t + t2 = 0 + 2 Combine like terms: 0 + 2 = 2 -2.5t + t2 = 2 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 = 2 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = 2 + 1.5625 Combine like terms: 2 + 1.5625 = 3.5625 1.5625 + -2.5t + t2 = 3.5625 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 3.5625 Calculate the square root of the right side: 1.887458609 Break this problem into two subproblems by setting (t + -1.25) equal to 1.887458609 and -1.887458609.Subproblem 1
t + -1.25 = 1.887458609 Simplifying t + -1.25 = 1.887458609 Reorder the terms: -1.25 + t = 1.887458609 Solving -1.25 + t = 1.887458609 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 = 1.887458609 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 1.887458609 + 1.25 t = 1.887458609 + 1.25 Combine like terms: 1.887458609 + 1.25 = 3.137458609 t = 3.137458609 Simplifying t = 3.137458609Subproblem 2
t + -1.25 = -1.887458609 Simplifying t + -1.25 = -1.887458609 Reorder the terms: -1.25 + t = -1.887458609 Solving -1.25 + t = -1.887458609 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 = -1.887458609 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -1.887458609 + 1.25 t = -1.887458609 + 1.25 Combine like terms: -1.887458609 + 1.25 = -0.637458609 t = -0.637458609 Simplifying t = -0.637458609Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.137458609, -0.637458609}Solution
t = {3.137458609, -0.637458609}
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