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Simplifying -8t * 3t + 2 = -5t + -6t Reorder the terms for easier multiplication: -8 * 3t * t + 2 = -5t + -6t Multiply -8 * 3 -24t * t + 2 = -5t + -6t Multiply t * t -24t2 + 2 = -5t + -6t Reorder the terms: 2 + -24t2 = -5t + -6t Combine like terms: -5t + -6t = -11t 2 + -24t2 = -11t Solving 2 + -24t2 = -11t Solving for variable 't'. Reorder the terms: 2 + 11t + -24t2 = -11t + 11t Combine like terms: -11t + 11t = 0 2 + 11t + -24t2 = 0 Begin completing the square. Divide all terms by -24 the coefficient of the squared term: Divide each side by '-24'. -0.08333333333 + -0.4583333333t + t2 = 0 Move the constant term to the right: Add '0.08333333333' to each side of the equation. -0.08333333333 + -0.4583333333t + 0.08333333333 + t2 = 0 + 0.08333333333 Reorder the terms: -0.08333333333 + 0.08333333333 + -0.4583333333t + t2 = 0 + 0.08333333333 Combine like terms: -0.08333333333 + 0.08333333333 = 0.00000000000 0.00000000000 + -0.4583333333t + t2 = 0 + 0.08333333333 -0.4583333333t + t2 = 0 + 0.08333333333 Combine like terms: 0 + 0.08333333333 = 0.08333333333 -0.4583333333t + t2 = 0.08333333333 The t term is -0.4583333333t. Take half its coefficient (-0.2291666667). Square it (0.05251736113) and add it to both sides. Add '0.05251736113' to each side of the equation. -0.4583333333t + 0.05251736113 + t2 = 0.08333333333 + 0.05251736113 Reorder the terms: 0.05251736113 + -0.4583333333t + t2 = 0.08333333333 + 0.05251736113 Combine like terms: 0.08333333333 + 0.05251736113 = 0.13585069446 0.05251736113 + -0.4583333333t + t2 = 0.13585069446 Factor a perfect square on the left side: (t + -0.2291666667)(t + -0.2291666667) = 0.13585069446 Calculate the square root of the right side: 0.368579292 Break this problem into two subproblems by setting (t + -0.2291666667) equal to 0.368579292 and -0.368579292.Subproblem 1
t + -0.2291666667 = 0.368579292 Simplifying t + -0.2291666667 = 0.368579292 Reorder the terms: -0.2291666667 + t = 0.368579292 Solving -0.2291666667 + t = 0.368579292 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.2291666667' to each side of the equation. -0.2291666667 + 0.2291666667 + t = 0.368579292 + 0.2291666667 Combine like terms: -0.2291666667 + 0.2291666667 = 0.0000000000 0.0000000000 + t = 0.368579292 + 0.2291666667 t = 0.368579292 + 0.2291666667 Combine like terms: 0.368579292 + 0.2291666667 = 0.5977459587 t = 0.5977459587 Simplifying t = 0.5977459587Subproblem 2
t + -0.2291666667 = -0.368579292 Simplifying t + -0.2291666667 = -0.368579292 Reorder the terms: -0.2291666667 + t = -0.368579292 Solving -0.2291666667 + t = -0.368579292 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.2291666667' to each side of the equation. -0.2291666667 + 0.2291666667 + t = -0.368579292 + 0.2291666667 Combine like terms: -0.2291666667 + 0.2291666667 = 0.0000000000 0.0000000000 + t = -0.368579292 + 0.2291666667 t = -0.368579292 + 0.2291666667 Combine like terms: -0.368579292 + 0.2291666667 = -0.1394126253 t = -0.1394126253 Simplifying t = -0.1394126253Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.5977459587, -0.1394126253}
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