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Simplifying -16t2 + 144t + -160 = 0 Reorder the terms: -160 + 144t + -16t2 = 0 Solving -160 + 144t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '16'. 16(-10 + 9t + -1t2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(-10 + 9t + -1t2)' equal to zero and attempt to solve: Simplifying -10 + 9t + -1t2 = 0 Solving -10 + 9t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 10 + -9t + t2 = 0 Move the constant term to the right: Add '-10' to each side of the equation. 10 + -9t + -10 + t2 = 0 + -10 Reorder the terms: 10 + -10 + -9t + t2 = 0 + -10 Combine like terms: 10 + -10 = 0 0 + -9t + t2 = 0 + -10 -9t + t2 = 0 + -10 Combine like terms: 0 + -10 = -10 -9t + t2 = -10 The t term is -9t. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9t + 20.25 + t2 = -10 + 20.25 Reorder the terms: 20.25 + -9t + t2 = -10 + 20.25 Combine like terms: -10 + 20.25 = 10.25 20.25 + -9t + t2 = 10.25 Factor a perfect square on the left side: (t + -4.5)(t + -4.5) = 10.25 Calculate the square root of the right side: 3.201562119 Break this problem into two subproblems by setting (t + -4.5) equal to 3.201562119 and -3.201562119.Subproblem 1
t + -4.5 = 3.201562119 Simplifying t + -4.5 = 3.201562119 Reorder the terms: -4.5 + t = 3.201562119 Solving -4.5 + t = 3.201562119 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = 3.201562119 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = 3.201562119 + 4.5 t = 3.201562119 + 4.5 Combine like terms: 3.201562119 + 4.5 = 7.701562119 t = 7.701562119 Simplifying t = 7.701562119Subproblem 2
t + -4.5 = -3.201562119 Simplifying t + -4.5 = -3.201562119 Reorder the terms: -4.5 + t = -3.201562119 Solving -4.5 + t = -3.201562119 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + t = -3.201562119 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + t = -3.201562119 + 4.5 t = -3.201562119 + 4.5 Combine like terms: -3.201562119 + 4.5 = 1.298437881 t = 1.298437881 Simplifying t = 1.298437881Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.701562119, 1.298437881}Solution
t = {7.701562119, 1.298437881}
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