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Simplifying 16t2 + 46t + 9 = 0 Reorder the terms: 9 + 46t + 16t2 = 0 Solving 9 + 46t + 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'. 0.5625 + 2.875t + t2 = 0 Move the constant term to the right: Add '-0.5625' to each side of the equation. 0.5625 + 2.875t + -0.5625 + t2 = 0 + -0.5625 Reorder the terms: 0.5625 + -0.5625 + 2.875t + t2 = 0 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + 2.875t + t2 = 0 + -0.5625 2.875t + t2 = 0 + -0.5625 Combine like terms: 0 + -0.5625 = -0.5625 2.875t + t2 = -0.5625 The t term is 2.875t. Take half its coefficient (1.4375). Square it (2.06640625) and add it to both sides. Add '2.06640625' to each side of the equation. 2.875t + 2.06640625 + t2 = -0.5625 + 2.06640625 Reorder the terms: 2.06640625 + 2.875t + t2 = -0.5625 + 2.06640625 Combine like terms: -0.5625 + 2.06640625 = 1.50390625 2.06640625 + 2.875t + t2 = 1.50390625 Factor a perfect square on the left side: (t + 1.4375)(t + 1.4375) = 1.50390625 Calculate the square root of the right side: 1.226338554 Break this problem into two subproblems by setting (t + 1.4375) equal to 1.226338554 and -1.226338554.Subproblem 1
t + 1.4375 = 1.226338554 Simplifying t + 1.4375 = 1.226338554 Reorder the terms: 1.4375 + t = 1.226338554 Solving 1.4375 + t = 1.226338554 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.4375' to each side of the equation. 1.4375 + -1.4375 + t = 1.226338554 + -1.4375 Combine like terms: 1.4375 + -1.4375 = 0.0000 0.0000 + t = 1.226338554 + -1.4375 t = 1.226338554 + -1.4375 Combine like terms: 1.226338554 + -1.4375 = -0.211161446 t = -0.211161446 Simplifying t = -0.211161446Subproblem 2
t + 1.4375 = -1.226338554 Simplifying t + 1.4375 = -1.226338554 Reorder the terms: 1.4375 + t = -1.226338554 Solving 1.4375 + t = -1.226338554 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.4375' to each side of the equation. 1.4375 + -1.4375 + t = -1.226338554 + -1.4375 Combine like terms: 1.4375 + -1.4375 = 0.0000 0.0000 + t = -1.226338554 + -1.4375 t = -1.226338554 + -1.4375 Combine like terms: -1.226338554 + -1.4375 = -2.663838554 t = -2.663838554 Simplifying t = -2.663838554Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.211161446, -2.663838554}
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