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Simplifying -20 = -16t2 + 32t + 20 Reorder the terms: -20 = 20 + 32t + -16t2 Solving -20 = 20 + 32t + -16t2 Solving for variable 't'. Combine like terms: -20 + -20 = -40 -40 + -32t + 16t2 = 20 + 32t + -16t2 + -20 + -32t + 16t2 Reorder the terms: -40 + -32t + 16t2 = 20 + -20 + 32t + -32t + -16t2 + 16t2 Combine like terms: 20 + -20 = 0 -40 + -32t + 16t2 = 0 + 32t + -32t + -16t2 + 16t2 -40 + -32t + 16t2 = 32t + -32t + -16t2 + 16t2 Combine like terms: 32t + -32t = 0 -40 + -32t + 16t2 = 0 + -16t2 + 16t2 -40 + -32t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -40 + -32t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-5 + -4t + 2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-5 + -4t + 2t2)' equal to zero and attempt to solve: Simplifying -5 + -4t + 2t2 = 0 Solving -5 + -4t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2.5 + -2t + t2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -2t + 2.5 + t2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -2t + t2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -2t + t2 = 0 + 2.5 -2t + t2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -2t + t2 = 2.5 The t term is -2t. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2t + 1 + t2 = 2.5 + 1 Reorder the terms: 1 + -2t + t2 = 2.5 + 1 Combine like terms: 2.5 + 1 = 3.5 1 + -2t + t2 = 3.5 Factor a perfect square on the left side: (t + -1)(t + -1) = 3.5 Calculate the square root of the right side: 1.870828693 Break this problem into two subproblems by setting (t + -1) equal to 1.870828693 and -1.870828693.Subproblem 1
t + -1 = 1.870828693 Simplifying t + -1 = 1.870828693 Reorder the terms: -1 + t = 1.870828693 Solving -1 + t = 1.870828693 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = 1.870828693 + 1 Combine like terms: -1 + 1 = 0 0 + t = 1.870828693 + 1 t = 1.870828693 + 1 Combine like terms: 1.870828693 + 1 = 2.870828693 t = 2.870828693 Simplifying t = 2.870828693Subproblem 2
t + -1 = -1.870828693 Simplifying t + -1 = -1.870828693 Reorder the terms: -1 + t = -1.870828693 Solving -1 + t = -1.870828693 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = -1.870828693 + 1 Combine like terms: -1 + 1 = 0 0 + t = -1.870828693 + 1 t = -1.870828693 + 1 Combine like terms: -1.870828693 + 1 = -0.870828693 t = -0.870828693 Simplifying t = -0.870828693Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.870828693, -0.870828693}Solution
t = {2.870828693, -0.870828693}
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