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Simplifying 5t2 + 15t + 2 = 0 Reorder the terms: 2 + 15t + 5t2 = 0 Solving 2 + 15t + 5t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.4 + 3t + t2 = 0 Move the constant term to the right: Add '-0.4' to each side of the equation. 0.4 + 3t + -0.4 + t2 = 0 + -0.4 Reorder the terms: 0.4 + -0.4 + 3t + t2 = 0 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + 3t + t2 = 0 + -0.4 3t + t2 = 0 + -0.4 Combine like terms: 0 + -0.4 = -0.4 3t + t2 = -0.4 The t term is 3t. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3t + 2.25 + t2 = -0.4 + 2.25 Reorder the terms: 2.25 + 3t + t2 = -0.4 + 2.25 Combine like terms: -0.4 + 2.25 = 1.85 2.25 + 3t + t2 = 1.85 Factor a perfect square on the left side: (t + 1.5)(t + 1.5) = 1.85 Calculate the square root of the right side: 1.360147051 Break this problem into two subproblems by setting (t + 1.5) equal to 1.360147051 and -1.360147051.Subproblem 1
t + 1.5 = 1.360147051 Simplifying t + 1.5 = 1.360147051 Reorder the terms: 1.5 + t = 1.360147051 Solving 1.5 + t = 1.360147051 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = 1.360147051 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = 1.360147051 + -1.5 t = 1.360147051 + -1.5 Combine like terms: 1.360147051 + -1.5 = -0.139852949 t = -0.139852949 Simplifying t = -0.139852949Subproblem 2
t + 1.5 = -1.360147051 Simplifying t + 1.5 = -1.360147051 Reorder the terms: 1.5 + t = -1.360147051 Solving 1.5 + t = -1.360147051 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = -1.360147051 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = -1.360147051 + -1.5 t = -1.360147051 + -1.5 Combine like terms: -1.360147051 + -1.5 = -2.860147051 t = -2.860147051 Simplifying t = -2.860147051Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.139852949, -2.860147051}
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