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Simplifying 20 = -16t2 + 16t + 36 Reorder the terms: 20 = 36 + 16t + -16t2 Solving 20 = 36 + 16t + -16t2 Solving for variable 't'. Combine like terms: 20 + -36 = -16 -16 + -16t + 16t2 = 36 + 16t + -16t2 + -36 + -16t + 16t2 Reorder the terms: -16 + -16t + 16t2 = 36 + -36 + 16t + -16t + -16t2 + 16t2 Combine like terms: 36 + -36 = 0 -16 + -16t + 16t2 = 0 + 16t + -16t + -16t2 + 16t2 -16 + -16t + 16t2 = 16t + -16t + -16t2 + 16t2 Combine like terms: 16t + -16t = 0 -16 + -16t + 16t2 = 0 + -16t2 + 16t2 -16 + -16t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -16 + -16t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '16'. 16(-1 + -1t + t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(-1 + -1t + t2)' equal to zero and attempt to solve: Simplifying -1 + -1t + t2 = 0 Solving -1 + -1t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + -1t + 1 + t2 = 0 + 1 Reorder the terms: -1 + 1 + -1t + t2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1t + t2 = 0 + 1 -1t + t2 = 0 + 1 Combine like terms: 0 + 1 = 1 -1t + t2 = 1 The t term is -1t. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1t + 0.25 + t2 = 1 + 0.25 Reorder the terms: 0.25 + -1t + t2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 0.25 + -1t + t2 = 1.25 Factor a perfect square on the left side: (t + -0.5)(t + -0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (t + -0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
t + -0.5 = 1.118033989 Simplifying t + -0.5 = 1.118033989 Reorder the terms: -0.5 + t = 1.118033989 Solving -0.5 + t = 1.118033989 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + t = 1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = 1.118033989 + 0.5 t = 1.118033989 + 0.5 Combine like terms: 1.118033989 + 0.5 = 1.618033989 t = 1.618033989 Simplifying t = 1.618033989Subproblem 2
t + -0.5 = -1.118033989 Simplifying t + -0.5 = -1.118033989 Reorder the terms: -0.5 + t = -1.118033989 Solving -0.5 + t = -1.118033989 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + t = -1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + t = -1.118033989 + 0.5 t = -1.118033989 + 0.5 Combine like terms: -1.118033989 + 0.5 = -0.618033989 t = -0.618033989 Simplifying t = -0.618033989Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.618033989, -0.618033989}Solution
t = {1.618033989, -0.618033989}
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