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Simplifying -16t2 + -13t + 120 = 0 Reorder the terms: 120 + -13t + -16t2 = 0 Solving 120 + -13t + -16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -7.5 + 0.8125t + t2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 0.8125t + 7.5 + t2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 0.8125t + t2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 0.8125t + t2 = 0 + 7.5 0.8125t + t2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 0.8125t + t2 = 7.5 The t term is 0.8125t. Take half its coefficient (0.40625). Square it (0.1650390625) and add it to both sides. Add '0.1650390625' to each side of the equation. 0.8125t + 0.1650390625 + t2 = 7.5 + 0.1650390625 Reorder the terms: 0.1650390625 + 0.8125t + t2 = 7.5 + 0.1650390625 Combine like terms: 7.5 + 0.1650390625 = 7.6650390625 0.1650390625 + 0.8125t + t2 = 7.6650390625 Factor a perfect square on the left side: (t + 0.40625)(t + 0.40625) = 7.6650390625 Calculate the square root of the right side: 2.768580695 Break this problem into two subproblems by setting (t + 0.40625) equal to 2.768580695 and -2.768580695.Subproblem 1
t + 0.40625 = 2.768580695 Simplifying t + 0.40625 = 2.768580695 Reorder the terms: 0.40625 + t = 2.768580695 Solving 0.40625 + t = 2.768580695 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.40625' to each side of the equation. 0.40625 + -0.40625 + t = 2.768580695 + -0.40625 Combine like terms: 0.40625 + -0.40625 = 0.00000 0.00000 + t = 2.768580695 + -0.40625 t = 2.768580695 + -0.40625 Combine like terms: 2.768580695 + -0.40625 = 2.362330695 t = 2.362330695 Simplifying t = 2.362330695Subproblem 2
t + 0.40625 = -2.768580695 Simplifying t + 0.40625 = -2.768580695 Reorder the terms: 0.40625 + t = -2.768580695 Solving 0.40625 + t = -2.768580695 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.40625' to each side of the equation. 0.40625 + -0.40625 + t = -2.768580695 + -0.40625 Combine like terms: 0.40625 + -0.40625 = 0.00000 0.00000 + t = -2.768580695 + -0.40625 t = -2.768580695 + -0.40625 Combine like terms: -2.768580695 + -0.40625 = -3.174830695 t = -3.174830695 Simplifying t = -3.174830695Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.362330695, -3.174830695}
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