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Simplifying 7.5 = -16t2 + 19t + 6 Reorder the terms: 7.5 = 6 + 19t + -16t2 Solving 7.5 = 6 + 19t + -16t2 Solving for variable 't'. Combine like terms: 7.5 + -6 = 1.5 1.5 + -19t + 16t2 = 6 + 19t + -16t2 + -6 + -19t + 16t2 Reorder the terms: 1.5 + -19t + 16t2 = 6 + -6 + 19t + -19t + -16t2 + 16t2 Combine like terms: 6 + -6 = 0 1.5 + -19t + 16t2 = 0 + 19t + -19t + -16t2 + 16t2 1.5 + -19t + 16t2 = 19t + -19t + -16t2 + 16t2 Combine like terms: 19t + -19t = 0 1.5 + -19t + 16t2 = 0 + -16t2 + 16t2 1.5 + -19t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 1.5 + -19t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.09375 + -1.1875t + t2 = 0 Move the constant term to the right: Add '-0.09375' to each side of the equation. 0.09375 + -1.1875t + -0.09375 + t2 = 0 + -0.09375 Reorder the terms: 0.09375 + -0.09375 + -1.1875t + t2 = 0 + -0.09375 Combine like terms: 0.09375 + -0.09375 = 0.00000 0.00000 + -1.1875t + t2 = 0 + -0.09375 -1.1875t + t2 = 0 + -0.09375 Combine like terms: 0 + -0.09375 = -0.09375 -1.1875t + t2 = -0.09375 The t term is -1.1875t. Take half its coefficient (-0.59375). Square it (0.3525390625) and add it to both sides. Add '0.3525390625' to each side of the equation. -1.1875t + 0.3525390625 + t2 = -0.09375 + 0.3525390625 Reorder the terms: 0.3525390625 + -1.1875t + t2 = -0.09375 + 0.3525390625 Combine like terms: -0.09375 + 0.3525390625 = 0.2587890625 0.3525390625 + -1.1875t + t2 = 0.2587890625 Factor a perfect square on the left side: (t + -0.59375)(t + -0.59375) = 0.2587890625 Calculate the square root of the right side: 0.508713144 Break this problem into two subproblems by setting (t + -0.59375) equal to 0.508713144 and -0.508713144.Subproblem 1
t + -0.59375 = 0.508713144 Simplifying t + -0.59375 = 0.508713144 Reorder the terms: -0.59375 + t = 0.508713144 Solving -0.59375 + t = 0.508713144 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.59375' to each side of the equation. -0.59375 + 0.59375 + t = 0.508713144 + 0.59375 Combine like terms: -0.59375 + 0.59375 = 0.00000 0.00000 + t = 0.508713144 + 0.59375 t = 0.508713144 + 0.59375 Combine like terms: 0.508713144 + 0.59375 = 1.102463144 t = 1.102463144 Simplifying t = 1.102463144Subproblem 2
t + -0.59375 = -0.508713144 Simplifying t + -0.59375 = -0.508713144 Reorder the terms: -0.59375 + t = -0.508713144 Solving -0.59375 + t = -0.508713144 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.59375' to each side of the equation. -0.59375 + 0.59375 + t = -0.508713144 + 0.59375 Combine like terms: -0.59375 + 0.59375 = 0.00000 0.00000 + t = -0.508713144 + 0.59375 t = -0.508713144 + 0.59375 Combine like terms: -0.508713144 + 0.59375 = 0.085036856 t = 0.085036856 Simplifying t = 0.085036856Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.102463144, 0.085036856}
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