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Simplifying 0.5 = -16t2 + 32t + 7 Reorder the terms: 0.5 = 7 + 32t + -16t2 Solving 0.5 = 7 + 32t + -16t2 Solving for variable 't'. Combine like terms: 0.5 + -7 = -6.5 -6.5 + -32t + 16t2 = 7 + 32t + -16t2 + -7 + -32t + 16t2 Reorder the terms: -6.5 + -32t + 16t2 = 7 + -7 + 32t + -32t + -16t2 + 16t2 Combine like terms: 7 + -7 = 0 -6.5 + -32t + 16t2 = 0 + 32t + -32t + -16t2 + 16t2 -6.5 + -32t + 16t2 = 32t + -32t + -16t2 + 16t2 Combine like terms: 32t + -32t = 0 -6.5 + -32t + 16t2 = 0 + -16t2 + 16t2 -6.5 + -32t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -6.5 + -32t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.40625 + -2t + t2 = 0 Move the constant term to the right: Add '0.40625' to each side of the equation. -0.40625 + -2t + 0.40625 + t2 = 0 + 0.40625 Reorder the terms: -0.40625 + 0.40625 + -2t + t2 = 0 + 0.40625 Combine like terms: -0.40625 + 0.40625 = 0.00000 0.00000 + -2t + t2 = 0 + 0.40625 -2t + t2 = 0 + 0.40625 Combine like terms: 0 + 0.40625 = 0.40625 -2t + t2 = 0.40625 The t term is -2t. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2t + 1 + t2 = 0.40625 + 1 Reorder the terms: 1 + -2t + t2 = 0.40625 + 1 Combine like terms: 0.40625 + 1 = 1.40625 1 + -2t + t2 = 1.40625 Factor a perfect square on the left side: (t + -1)(t + -1) = 1.40625 Calculate the square root of the right side: 1.185854123 Break this problem into two subproblems by setting (t + -1) equal to 1.185854123 and -1.185854123.Subproblem 1
t + -1 = 1.185854123 Simplifying t + -1 = 1.185854123 Reorder the terms: -1 + t = 1.185854123 Solving -1 + t = 1.185854123 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = 1.185854123 + 1 Combine like terms: -1 + 1 = 0 0 + t = 1.185854123 + 1 t = 1.185854123 + 1 Combine like terms: 1.185854123 + 1 = 2.185854123 t = 2.185854123 Simplifying t = 2.185854123Subproblem 2
t + -1 = -1.185854123 Simplifying t + -1 = -1.185854123 Reorder the terms: -1 + t = -1.185854123 Solving -1 + t = -1.185854123 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + t = -1.185854123 + 1 Combine like terms: -1 + 1 = 0 0 + t = -1.185854123 + 1 t = -1.185854123 + 1 Combine like terms: -1.185854123 + 1 = -0.185854123 t = -0.185854123 Simplifying t = -0.185854123Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.185854123, -0.185854123}
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