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Simplifying 16t2 + 24t + 1 = 0 Reorder the terms: 1 + 24t + 16t2 = 0 Solving 1 + 24t + 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'. 0.0625 + 1.5t + t2 = 0 Move the constant term to the right: Add '-0.0625' to each side of the equation. 0.0625 + 1.5t + -0.0625 + t2 = 0 + -0.0625 Reorder the terms: 0.0625 + -0.0625 + 1.5t + t2 = 0 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + 1.5t + t2 = 0 + -0.0625 1.5t + t2 = 0 + -0.0625 Combine like terms: 0 + -0.0625 = -0.0625 1.5t + t2 = -0.0625 The t term is 1.5t. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5t + 0.5625 + t2 = -0.0625 + 0.5625 Reorder the terms: 0.5625 + 1.5t + t2 = -0.0625 + 0.5625 Combine like terms: -0.0625 + 0.5625 = 0.5 0.5625 + 1.5t + t2 = 0.5 Factor a perfect square on the left side: (t + 0.75)(t + 0.75) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (t + 0.75) equal to 0.707106781 and -0.707106781.Subproblem 1
t + 0.75 = 0.707106781 Simplifying t + 0.75 = 0.707106781 Reorder the terms: 0.75 + t = 0.707106781 Solving 0.75 + t = 0.707106781 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + t = 0.707106781 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = 0.707106781 + -0.75 t = 0.707106781 + -0.75 Combine like terms: 0.707106781 + -0.75 = -0.042893219 t = -0.042893219 Simplifying t = -0.042893219Subproblem 2
t + 0.75 = -0.707106781 Simplifying t + 0.75 = -0.707106781 Reorder the terms: 0.75 + t = -0.707106781 Solving 0.75 + t = -0.707106781 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + t = -0.707106781 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = -0.707106781 + -0.75 t = -0.707106781 + -0.75 Combine like terms: -0.707106781 + -0.75 = -1.457106781 t = -1.457106781 Simplifying t = -1.457106781Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.042893219, -1.457106781}
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