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Simplifying 40 = 112t + -16t2 Solving 40 = 112t + -16t2 Solving for variable 't'. Reorder the terms: 40 + -112t + 16t2 = 112t + -112t + -16t2 + 16t2 Combine like terms: 112t + -112t = 0 40 + -112t + 16t2 = 0 + -16t2 + 16t2 40 + -112t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 40 + -112t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(5 + -14t + 2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(5 + -14t + 2t2)' equal to zero and attempt to solve: Simplifying 5 + -14t + 2t2 = 0 Solving 5 + -14t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 2.5 + -7t + t2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + -7t + -2.5 + t2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + -7t + t2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + -7t + t2 = 0 + -2.5 -7t + t2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 -7t + t2 = -2.5 The t term is -7t. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7t + 12.25 + t2 = -2.5 + 12.25 Reorder the terms: 12.25 + -7t + t2 = -2.5 + 12.25 Combine like terms: -2.5 + 12.25 = 9.75 12.25 + -7t + t2 = 9.75 Factor a perfect square on the left side: (t + -3.5)(t + -3.5) = 9.75 Calculate the square root of the right side: 3.122498999 Break this problem into two subproblems by setting (t + -3.5) equal to 3.122498999 and -3.122498999.Subproblem 1
t + -3.5 = 3.122498999 Simplifying t + -3.5 = 3.122498999 Reorder the terms: -3.5 + t = 3.122498999 Solving -3.5 + t = 3.122498999 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + t = 3.122498999 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + t = 3.122498999 + 3.5 t = 3.122498999 + 3.5 Combine like terms: 3.122498999 + 3.5 = 6.622498999 t = 6.622498999 Simplifying t = 6.622498999Subproblem 2
t + -3.5 = -3.122498999 Simplifying t + -3.5 = -3.122498999 Reorder the terms: -3.5 + t = -3.122498999 Solving -3.5 + t = -3.122498999 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + t = -3.122498999 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + t = -3.122498999 + 3.5 t = -3.122498999 + 3.5 Combine like terms: -3.122498999 + 3.5 = 0.377501001 t = 0.377501001 Simplifying t = 0.377501001Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.622498999, 0.377501001}Solution
t = {6.622498999, 0.377501001}
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