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Simplifying 2t2 + 15t + 20 = 0 Reorder the terms: 20 + 15t + 2t2 = 0 Solving 20 + 15t + 2t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 10 + 7.5t + t2 = 0 Move the constant term to the right: Add '-10' to each side of the equation. 10 + 7.5t + -10 + t2 = 0 + -10 Reorder the terms: 10 + -10 + 7.5t + t2 = 0 + -10 Combine like terms: 10 + -10 = 0 0 + 7.5t + t2 = 0 + -10 7.5t + t2 = 0 + -10 Combine like terms: 0 + -10 = -10 7.5t + t2 = -10 The t term is 7.5t. Take half its coefficient (3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. 7.5t + 14.0625 + t2 = -10 + 14.0625 Reorder the terms: 14.0625 + 7.5t + t2 = -10 + 14.0625 Combine like terms: -10 + 14.0625 = 4.0625 14.0625 + 7.5t + t2 = 4.0625 Factor a perfect square on the left side: (t + 3.75)(t + 3.75) = 4.0625 Calculate the square root of the right side: 2.015564437 Break this problem into two subproblems by setting (t + 3.75) equal to 2.015564437 and -2.015564437.Subproblem 1
t + 3.75 = 2.015564437 Simplifying t + 3.75 = 2.015564437 Reorder the terms: 3.75 + t = 2.015564437 Solving 3.75 + t = 2.015564437 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + t = 2.015564437 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + t = 2.015564437 + -3.75 t = 2.015564437 + -3.75 Combine like terms: 2.015564437 + -3.75 = -1.734435563 t = -1.734435563 Simplifying t = -1.734435563Subproblem 2
t + 3.75 = -2.015564437 Simplifying t + 3.75 = -2.015564437 Reorder the terms: 3.75 + t = -2.015564437 Solving 3.75 + t = -2.015564437 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + t = -2.015564437 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + t = -2.015564437 + -3.75 t = -2.015564437 + -3.75 Combine like terms: -2.015564437 + -3.75 = -5.765564437 t = -5.765564437 Simplifying t = -5.765564437Solution
The solution to the problem is based on the solutions from the subproblems. t = {-1.734435563, -5.765564437}
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