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Simplifying 8 = -16t2 + 34t + -4 Reorder the terms: 8 = -4 + 34t + -16t2 Solving 8 = -4 + 34t + -16t2 Solving for variable 't'. Combine like terms: 8 + 4 = 12 12 + -34t + 16t2 = -4 + 34t + -16t2 + 4 + -34t + 16t2 Reorder the terms: 12 + -34t + 16t2 = -4 + 4 + 34t + -34t + -16t2 + 16t2 Combine like terms: -4 + 4 = 0 12 + -34t + 16t2 = 0 + 34t + -34t + -16t2 + 16t2 12 + -34t + 16t2 = 34t + -34t + -16t2 + 16t2 Combine like terms: 34t + -34t = 0 12 + -34t + 16t2 = 0 + -16t2 + 16t2 12 + -34t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 12 + -34t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(6 + -17t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(6 + -17t + 8t2)' equal to zero and attempt to solve: Simplifying 6 + -17t + 8t2 = 0 Solving 6 + -17t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.75 + -2.125t + t2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + -2.125t + -0.75 + t2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + -2.125t + t2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + -2.125t + t2 = 0 + -0.75 -2.125t + t2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 -2.125t + t2 = -0.75 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 = -0.75 + 1.12890625 Reorder the terms: 1.12890625 + -2.125t + t2 = -0.75 + 1.12890625 Combine like terms: -0.75 + 1.12890625 = 0.37890625 1.12890625 + -2.125t + t2 = 0.37890625 Factor a perfect square on the left side: (t + -1.0625)(t + -1.0625) = 0.37890625 Calculate the square root of the right side: 0.615553613 Break this problem into two subproblems by setting (t + -1.0625) equal to 0.615553613 and -0.615553613.Subproblem 1
t + -1.0625 = 0.615553613 Simplifying t + -1.0625 = 0.615553613 Reorder the terms: -1.0625 + t = 0.615553613 Solving -1.0625 + t = 0.615553613 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 = 0.615553613 + 1.0625 Combine like terms: -1.0625 + 1.0625 = 0.0000 0.0000 + t = 0.615553613 + 1.0625 t = 0.615553613 + 1.0625 Combine like terms: 0.615553613 + 1.0625 = 1.678053613 t = 1.678053613 Simplifying t = 1.678053613Subproblem 2
t + -1.0625 = -0.615553613 Simplifying t + -1.0625 = -0.615553613 Reorder the terms: -1.0625 + t = -0.615553613 Solving -1.0625 + t = -0.615553613 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 = -0.615553613 + 1.0625 Combine like terms: -1.0625 + 1.0625 = 0.0000 0.0000 + t = -0.615553613 + 1.0625 t = -0.615553613 + 1.0625 Combine like terms: -0.615553613 + 1.0625 = 0.446946387 t = 0.446946387 Simplifying t = 0.446946387Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.678053613, 0.446946387}Solution
t = {1.678053613, 0.446946387}
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