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Simplifying 16t2 + 25t + 3 = 0 Reorder the terms: 3 + 25t + 16t2 = 0 Solving 3 + 25t + 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'. 0.1875 + 1.5625t + t2 = 0 Move the constant term to the right: Add '-0.1875' to each side of the equation. 0.1875 + 1.5625t + -0.1875 + t2 = 0 + -0.1875 Reorder the terms: 0.1875 + -0.1875 + 1.5625t + t2 = 0 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + 1.5625t + t2 = 0 + -0.1875 1.5625t + t2 = 0 + -0.1875 Combine like terms: 0 + -0.1875 = -0.1875 1.5625t + t2 = -0.1875 The t term is 1.5625t. Take half its coefficient (0.78125). Square it (0.6103515625) and add it to both sides. Add '0.6103515625' to each side of the equation. 1.5625t + 0.6103515625 + t2 = -0.1875 + 0.6103515625 Reorder the terms: 0.6103515625 + 1.5625t + t2 = -0.1875 + 0.6103515625 Combine like terms: -0.1875 + 0.6103515625 = 0.4228515625 0.6103515625 + 1.5625t + t2 = 0.4228515625 Factor a perfect square on the left side: (t + 0.78125)(t + 0.78125) = 0.4228515625 Calculate the square root of the right side: 0.650270376 Break this problem into two subproblems by setting (t + 0.78125) equal to 0.650270376 and -0.650270376.Subproblem 1
t + 0.78125 = 0.650270376 Simplifying t + 0.78125 = 0.650270376 Reorder the terms: 0.78125 + t = 0.650270376 Solving 0.78125 + t = 0.650270376 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.78125' to each side of the equation. 0.78125 + -0.78125 + t = 0.650270376 + -0.78125 Combine like terms: 0.78125 + -0.78125 = 0.00000 0.00000 + t = 0.650270376 + -0.78125 t = 0.650270376 + -0.78125 Combine like terms: 0.650270376 + -0.78125 = -0.130979624 t = -0.130979624 Simplifying t = -0.130979624Subproblem 2
t + 0.78125 = -0.650270376 Simplifying t + 0.78125 = -0.650270376 Reorder the terms: 0.78125 + t = -0.650270376 Solving 0.78125 + t = -0.650270376 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.78125' to each side of the equation. 0.78125 + -0.78125 + t = -0.650270376 + -0.78125 Combine like terms: 0.78125 + -0.78125 = 0.00000 0.00000 + t = -0.650270376 + -0.78125 t = -0.650270376 + -0.78125 Combine like terms: -0.650270376 + -0.78125 = -1.431520376 t = -1.431520376 Simplifying t = -1.431520376Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.130979624, -1.431520376}
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