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Simplifying t2 + -10t = -23 Reorder the terms: -10t + t2 = -23 Solving -10t + t2 = -23 Solving for variable 't'. Reorder the terms: 23 + -10t + t2 = -23 + 23 Combine like terms: -23 + 23 = 0 23 + -10t + t2 = 0 Begin completing the square. Move the constant term to the right: Add '-23' to each side of the equation. 23 + -10t + -23 + t2 = 0 + -23 Reorder the terms: 23 + -23 + -10t + t2 = 0 + -23 Combine like terms: 23 + -23 = 0 0 + -10t + t2 = 0 + -23 -10t + t2 = 0 + -23 Combine like terms: 0 + -23 = -23 -10t + t2 = -23 The t term is -10t. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10t + 25 + t2 = -23 + 25 Reorder the terms: 25 + -10t + t2 = -23 + 25 Combine like terms: -23 + 25 = 2 25 + -10t + t2 = 2 Factor a perfect square on the left side: (t + -5)(t + -5) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (t + -5) equal to 1.414213562 and -1.414213562.Subproblem 1
t + -5 = 1.414213562 Simplifying t + -5 = 1.414213562 Reorder the terms: -5 + t = 1.414213562 Solving -5 + t = 1.414213562 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + t = 1.414213562 + 5 Combine like terms: -5 + 5 = 0 0 + t = 1.414213562 + 5 t = 1.414213562 + 5 Combine like terms: 1.414213562 + 5 = 6.414213562 t = 6.414213562 Simplifying t = 6.414213562Subproblem 2
t + -5 = -1.414213562 Simplifying t + -5 = -1.414213562 Reorder the terms: -5 + t = -1.414213562 Solving -5 + t = -1.414213562 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + t = -1.414213562 + 5 Combine like terms: -5 + 5 = 0 0 + t = -1.414213562 + 5 t = -1.414213562 + 5 Combine like terms: -1.414213562 + 5 = 3.585786438 t = 3.585786438 Simplifying t = 3.585786438Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.414213562, 3.585786438}
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