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Simplifying -24t2 + 15t + 5 = 3 Reorder the terms: 5 + 15t + -24t2 = 3 Solving 5 + 15t + -24t2 = 3 Solving for variable 't'. Reorder the terms: 5 + -3 + 15t + -24t2 = 3 + -3 Combine like terms: 5 + -3 = 2 2 + 15t + -24t2 = 3 + -3 Combine like terms: 3 + -3 = 0 2 + 15t + -24t2 = 0 Begin completing the square. Divide all terms by -24 the coefficient of the squared term: Divide each side by '-24'. -0.08333333333 + -0.625t + t2 = 0 Move the constant term to the right: Add '0.08333333333' to each side of the equation. -0.08333333333 + -0.625t + 0.08333333333 + t2 = 0 + 0.08333333333 Reorder the terms: -0.08333333333 + 0.08333333333 + -0.625t + t2 = 0 + 0.08333333333 Combine like terms: -0.08333333333 + 0.08333333333 = 0.00000000000 0.00000000000 + -0.625t + t2 = 0 + 0.08333333333 -0.625t + t2 = 0 + 0.08333333333 Combine like terms: 0 + 0.08333333333 = 0.08333333333 -0.625t + t2 = 0.08333333333 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 = 0.08333333333 + 0.09765625 Reorder the terms: 0.09765625 + -0.625t + t2 = 0.08333333333 + 0.09765625 Combine like terms: 0.08333333333 + 0.09765625 = 0.18098958333 0.09765625 + -0.625t + t2 = 0.18098958333 Factor a perfect square on the left side: (t + -0.3125)(t + -0.3125) = 0.18098958333 Calculate the square root of the right side: 0.425428705 Break this problem into two subproblems by setting (t + -0.3125) equal to 0.425428705 and -0.425428705.Subproblem 1
t + -0.3125 = 0.425428705 Simplifying t + -0.3125 = 0.425428705 Reorder the terms: -0.3125 + t = 0.425428705 Solving -0.3125 + t = 0.425428705 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 = 0.425428705 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + t = 0.425428705 + 0.3125 t = 0.425428705 + 0.3125 Combine like terms: 0.425428705 + 0.3125 = 0.737928705 t = 0.737928705 Simplifying t = 0.737928705Subproblem 2
t + -0.3125 = -0.425428705 Simplifying t + -0.3125 = -0.425428705 Reorder the terms: -0.3125 + t = -0.425428705 Solving -0.3125 + t = -0.425428705 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 = -0.425428705 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + t = -0.425428705 + 0.3125 t = -0.425428705 + 0.3125 Combine like terms: -0.425428705 + 0.3125 = -0.112928705 t = -0.112928705 Simplifying t = -0.112928705Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.737928705, -0.112928705}
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