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Simplifying -16t2 + 40t + 120 = 0 Reorder the terms: 120 + 40t + -16t2 = 0 Solving 120 + 40t + -16t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '8'. 8(15 + 5t + -2t2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(15 + 5t + -2t2)' equal to zero and attempt to solve: Simplifying 15 + 5t + -2t2 = 0 Solving 15 + 5t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -7.5 + -2.5t + t2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + -2.5t + 7.5 + t2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + -2.5t + t2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + -2.5t + t2 = 0 + 7.5 -2.5t + t2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 -2.5t + t2 = 7.5 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 = 7.5 + 1.5625 Reorder the terms: 1.5625 + -2.5t + t2 = 7.5 + 1.5625 Combine like terms: 7.5 + 1.5625 = 9.0625 1.5625 + -2.5t + t2 = 9.0625 Factor a perfect square on the left side: (t + -1.25)(t + -1.25) = 9.0625 Calculate the square root of the right side: 3.010398645 Break this problem into two subproblems by setting (t + -1.25) equal to 3.010398645 and -3.010398645.Subproblem 1
t + -1.25 = 3.010398645 Simplifying t + -1.25 = 3.010398645 Reorder the terms: -1.25 + t = 3.010398645 Solving -1.25 + t = 3.010398645 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 = 3.010398645 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = 3.010398645 + 1.25 t = 3.010398645 + 1.25 Combine like terms: 3.010398645 + 1.25 = 4.260398645 t = 4.260398645 Simplifying t = 4.260398645Subproblem 2
t + -1.25 = -3.010398645 Simplifying t + -1.25 = -3.010398645 Reorder the terms: -1.25 + t = -3.010398645 Solving -1.25 + t = -3.010398645 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 = -3.010398645 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + t = -3.010398645 + 1.25 t = -3.010398645 + 1.25 Combine like terms: -3.010398645 + 1.25 = -1.760398645 t = -1.760398645 Simplifying t = -1.760398645Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.260398645, -1.760398645}Solution
t = {4.260398645, -1.760398645}
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