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Simplifying 0 = 16t2 + 34t + 1.5 Reorder the terms: 0 = 1.5 + 34t + 16t2 Solving 0 = 1.5 + 34t + 16t2 Solving for variable 't'. Combine like terms: 0 + -1.5 = -1.5 -1.5 + -34t + -16t2 = 1.5 + 34t + 16t2 + -1.5 + -34t + -16t2 Reorder the terms: -1.5 + -34t + -16t2 = 1.5 + -1.5 + 34t + -34t + 16t2 + -16t2 Combine like terms: 1.5 + -1.5 = 0.0 -1.5 + -34t + -16t2 = 0.0 + 34t + -34t + 16t2 + -16t2 -1.5 + -34t + -16t2 = 34t + -34t + 16t2 + -16t2 Combine like terms: 34t + -34t = 0 -1.5 + -34t + -16t2 = 0 + 16t2 + -16t2 -1.5 + -34t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -1.5 + -34t + -16t2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.09375 + 2.125t + t2 = 0 Move the constant term to the right: Add '-0.09375' to each side of the equation. 0.09375 + 2.125t + -0.09375 + t2 = 0 + -0.09375 Reorder the terms: 0.09375 + -0.09375 + 2.125t + t2 = 0 + -0.09375 Combine like terms: 0.09375 + -0.09375 = 0.00000 0.00000 + 2.125t + t2 = 0 + -0.09375 2.125t + t2 = 0 + -0.09375 Combine like terms: 0 + -0.09375 = -0.09375 2.125t + t2 = -0.09375 The t term is 2.125t. Take half its coefficient (1.0625). Square it (1.12890625) and add it to both sides. Add '1.12890625' to each side of the equation. 2.125t + 1.12890625 + t2 = -0.09375 + 1.12890625 Reorder the terms: 1.12890625 + 2.125t + t2 = -0.09375 + 1.12890625 Combine like terms: -0.09375 + 1.12890625 = 1.03515625 1.12890625 + 2.125t + t2 = 1.03515625 Factor a perfect square on the left side: (t + 1.0625)(t + 1.0625) = 1.03515625 Calculate the square root of the right side: 1.017426287 Break this problem into two subproblems by setting (t + 1.0625) equal to 1.017426287 and -1.017426287.Subproblem 1
t + 1.0625 = 1.017426287 Simplifying t + 1.0625 = 1.017426287 Reorder the terms: 1.0625 + t = 1.017426287 Solving 1.0625 + t = 1.017426287 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.0625' to each side of the equation. 1.0625 + -1.0625 + t = 1.017426287 + -1.0625 Combine like terms: 1.0625 + -1.0625 = 0.0000 0.0000 + t = 1.017426287 + -1.0625 t = 1.017426287 + -1.0625 Combine like terms: 1.017426287 + -1.0625 = -0.045073713 t = -0.045073713 Simplifying t = -0.045073713Subproblem 2
t + 1.0625 = -1.017426287 Simplifying t + 1.0625 = -1.017426287 Reorder the terms: 1.0625 + t = -1.017426287 Solving 1.0625 + t = -1.017426287 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.0625' to each side of the equation. 1.0625 + -1.0625 + t = -1.017426287 + -1.0625 Combine like terms: 1.0625 + -1.0625 = 0.0000 0.0000 + t = -1.017426287 + -1.0625 t = -1.017426287 + -1.0625 Combine like terms: -1.017426287 + -1.0625 = -2.079926287 t = -2.079926287 Simplifying t = -2.079926287Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.045073713, -2.079926287}
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