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Simplifying 0 = -16t2 + 10t + 8 Reorder the terms: 0 = 8 + 10t + -16t2 Solving 0 = 8 + 10t + -16t2 Solving for variable 't'. Combine like terms: 0 + -8 = -8 -8 + -10t + 16t2 = 8 + 10t + -16t2 + -8 + -10t + 16t2 Reorder the terms: -8 + -10t + 16t2 = 8 + -8 + 10t + -10t + -16t2 + 16t2 Combine like terms: 8 + -8 = 0 -8 + -10t + 16t2 = 0 + 10t + -10t + -16t2 + 16t2 -8 + -10t + 16t2 = 10t + -10t + -16t2 + 16t2 Combine like terms: 10t + -10t = 0 -8 + -10t + 16t2 = 0 + -16t2 + 16t2 -8 + -10t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -8 + -10t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-4 + -5t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-4 + -5t + 8t2)' equal to zero and attempt to solve: Simplifying -4 + -5t + 8t2 = 0 Solving -4 + -5t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.5 + -0.625t + t2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -0.625t + 0.5 + t2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -0.625t + t2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -0.625t + t2 = 0 + 0.5 -0.625t + t2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -0.625t + t2 = 0.5 The t term is -0.625t. Take half its coefficient (-0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. -0.625t + 0.09765625 + t2 = 0.5 + 0.09765625 Reorder the terms: 0.09765625 + -0.625t + t2 = 0.5 + 0.09765625 Combine like terms: 0.5 + 0.09765625 = 0.59765625 0.09765625 + -0.625t + t2 = 0.59765625 Factor a perfect square on the left side: (t + -0.3125)(t + -0.3125) = 0.59765625 Calculate the square root of the right side: 0.773082305 Break this problem into two subproblems by setting (t + -0.3125) equal to 0.773082305 and -0.773082305.Subproblem 1
t + -0.3125 = 0.773082305 Simplifying t + -0.3125 = 0.773082305 Reorder the terms: -0.3125 + t = 0.773082305 Solving -0.3125 + t = 0.773082305 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + t = 0.773082305 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + t = 0.773082305 + 0.3125 t = 0.773082305 + 0.3125 Combine like terms: 0.773082305 + 0.3125 = 1.085582305 t = 1.085582305 Simplifying t = 1.085582305Subproblem 2
t + -0.3125 = -0.773082305 Simplifying t + -0.3125 = -0.773082305 Reorder the terms: -0.3125 + t = -0.773082305 Solving -0.3125 + t = -0.773082305 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + t = -0.773082305 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + t = -0.773082305 + 0.3125 t = -0.773082305 + 0.3125 Combine like terms: -0.773082305 + 0.3125 = -0.460582305 t = -0.460582305 Simplifying t = -0.460582305Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.085582305, -0.460582305}Solution
t = {1.085582305, -0.460582305}
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