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Simplifying 16t * 16t + 38.4t + 0.96 = 3 Reorder the terms for easier multiplication: 16 * 16t * t + 38.4t + 0.96 = 3 Multiply 16 * 16 256t * t + 38.4t + 0.96 = 3 Multiply t * t 256t2 + 38.4t + 0.96 = 3 Reorder the terms: 0.96 + 38.4t + 256t2 = 3 Solving 0.96 + 38.4t + 256t2 = 3 Solving for variable 't'. Reorder the terms: 0.96 + -3 + 38.4t + 256t2 = 3 + -3 Combine like terms: 0.96 + -3 = -2.04 -2.04 + 38.4t + 256t2 = 3 + -3 Combine like terms: 3 + -3 = 0 -2.04 + 38.4t + 256t2 = 0 Begin completing the square. Divide all terms by 256 the coefficient of the squared term: Divide each side by '256'. -0.00796875 + 0.15t + t2 = 0 Move the constant term to the right: Add '0.00796875' to each side of the equation. -0.00796875 + 0.15t + 0.00796875 + t2 = 0 + 0.00796875 Reorder the terms: -0.00796875 + 0.00796875 + 0.15t + t2 = 0 + 0.00796875 Combine like terms: -0.00796875 + 0.00796875 = 0.00000000 0.00000000 + 0.15t + t2 = 0 + 0.00796875 0.15t + t2 = 0 + 0.00796875 Combine like terms: 0 + 0.00796875 = 0.00796875 0.15t + t2 = 0.00796875 The t term is 0.15t. Take half its coefficient (0.075). Square it (0.005625) and add it to both sides. Add '0.005625' to each side of the equation. 0.15t + 0.005625 + t2 = 0.00796875 + 0.005625 Reorder the terms: 0.005625 + 0.15t + t2 = 0.00796875 + 0.005625 Combine like terms: 0.00796875 + 0.005625 = 0.01359375 0.005625 + 0.15t + t2 = 0.01359375 Factor a perfect square on the left side: (t + 0.075)(t + 0.075) = 0.01359375 Calculate the square root of the right side: 0.116592238 Break this problem into two subproblems by setting (t + 0.075) equal to 0.116592238 and -0.116592238.Subproblem 1
t + 0.075 = 0.116592238 Simplifying t + 0.075 = 0.116592238 Reorder the terms: 0.075 + t = 0.116592238 Solving 0.075 + t = 0.116592238 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.075' to each side of the equation. 0.075 + -0.075 + t = 0.116592238 + -0.075 Combine like terms: 0.075 + -0.075 = 0.000 0.000 + t = 0.116592238 + -0.075 t = 0.116592238 + -0.075 Combine like terms: 0.116592238 + -0.075 = 0.041592238 t = 0.041592238 Simplifying t = 0.041592238Subproblem 2
t + 0.075 = -0.116592238 Simplifying t + 0.075 = -0.116592238 Reorder the terms: 0.075 + t = -0.116592238 Solving 0.075 + t = -0.116592238 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.075' to each side of the equation. 0.075 + -0.075 + t = -0.116592238 + -0.075 Combine like terms: 0.075 + -0.075 = 0.000 0.000 + t = -0.116592238 + -0.075 t = -0.116592238 + -0.075 Combine like terms: -0.116592238 + -0.075 = -0.191592238 t = -0.191592238 Simplifying t = -0.191592238Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.041592238, -0.191592238}
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