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Simplifying -5t2 + 5t + 24 = t Reorder the terms: 24 + 5t + -5t2 = t Solving 24 + 5t + -5t2 = t Solving for variable 't'. Reorder the terms: 24 + 5t + -1t + -5t2 = t + -1t Combine like terms: 5t + -1t = 4t 24 + 4t + -5t2 = t + -1t Combine like terms: t + -1t = 0 24 + 4t + -5t2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -4.8 + -0.8t + t2 = 0 Move the constant term to the right: Add '4.8' to each side of the equation. -4.8 + -0.8t + 4.8 + t2 = 0 + 4.8 Reorder the terms: -4.8 + 4.8 + -0.8t + t2 = 0 + 4.8 Combine like terms: -4.8 + 4.8 = 0.0 0.0 + -0.8t + t2 = 0 + 4.8 -0.8t + t2 = 0 + 4.8 Combine like terms: 0 + 4.8 = 4.8 -0.8t + t2 = 4.8 The t term is -0.8t. Take half its coefficient (-0.4). Square it (0.16) and add it to both sides. Add '0.16' to each side of the equation. -0.8t + 0.16 + t2 = 4.8 + 0.16 Reorder the terms: 0.16 + -0.8t + t2 = 4.8 + 0.16 Combine like terms: 4.8 + 0.16 = 4.96 0.16 + -0.8t + t2 = 4.96 Factor a perfect square on the left side: (t + -0.4)(t + -0.4) = 4.96 Calculate the square root of the right side: 2.227105745 Break this problem into two subproblems by setting (t + -0.4) equal to 2.227105745 and -2.227105745.Subproblem 1
t + -0.4 = 2.227105745 Simplifying t + -0.4 = 2.227105745 Reorder the terms: -0.4 + t = 2.227105745 Solving -0.4 + t = 2.227105745 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + t = 2.227105745 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + t = 2.227105745 + 0.4 t = 2.227105745 + 0.4 Combine like terms: 2.227105745 + 0.4 = 2.627105745 t = 2.627105745 Simplifying t = 2.627105745Subproblem 2
t + -0.4 = -2.227105745 Simplifying t + -0.4 = -2.227105745 Reorder the terms: -0.4 + t = -2.227105745 Solving -0.4 + t = -2.227105745 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + t = -2.227105745 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + t = -2.227105745 + 0.4 t = -2.227105745 + 0.4 Combine like terms: -2.227105745 + 0.4 = -1.827105745 t = -1.827105745 Simplifying t = -1.827105745Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.627105745, -1.827105745}
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