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Simplifying 2t2 + -4.2t + 459 = 869 Reorder the terms: 459 + -4.2t + 2t2 = 869 Solving 459 + -4.2t + 2t2 = 869 Solving for variable 't'. Reorder the terms: 459 + -869 + -4.2t + 2t2 = 869 + -869 Combine like terms: 459 + -869 = -410 -410 + -4.2t + 2t2 = 869 + -869 Combine like terms: 869 + -869 = 0 -410 + -4.2t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -205 + -2.1t + t2 = 0 Move the constant term to the right: Add '205' to each side of the equation. -205 + -2.1t + 205 + t2 = 0 + 205 Reorder the terms: -205 + 205 + -2.1t + t2 = 0 + 205 Combine like terms: -205 + 205 = 0 0 + -2.1t + t2 = 0 + 205 -2.1t + t2 = 0 + 205 Combine like terms: 0 + 205 = 205 -2.1t + t2 = 205 The t term is -2.1t. Take half its coefficient (-1.05). Square it (1.1025) and add it to both sides. Add '1.1025' to each side of the equation. -2.1t + 1.1025 + t2 = 205 + 1.1025 Reorder the terms: 1.1025 + -2.1t + t2 = 205 + 1.1025 Combine like terms: 205 + 1.1025 = 206.1025 1.1025 + -2.1t + t2 = 206.1025 Factor a perfect square on the left side: (t + -1.05)(t + -1.05) = 206.1025 Calculate the square root of the right side: 14.356270407 Break this problem into two subproblems by setting (t + -1.05) equal to 14.356270407 and -14.356270407.Subproblem 1
t + -1.05 = 14.356270407 Simplifying t + -1.05 = 14.356270407 Reorder the terms: -1.05 + t = 14.356270407 Solving -1.05 + t = 14.356270407 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.05' to each side of the equation. -1.05 + 1.05 + t = 14.356270407 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + t = 14.356270407 + 1.05 t = 14.356270407 + 1.05 Combine like terms: 14.356270407 + 1.05 = 15.406270407 t = 15.406270407 Simplifying t = 15.406270407Subproblem 2
t + -1.05 = -14.356270407 Simplifying t + -1.05 = -14.356270407 Reorder the terms: -1.05 + t = -14.356270407 Solving -1.05 + t = -14.356270407 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.05' to each side of the equation. -1.05 + 1.05 + t = -14.356270407 + 1.05 Combine like terms: -1.05 + 1.05 = 0.00 0.00 + t = -14.356270407 + 1.05 t = -14.356270407 + 1.05 Combine like terms: -14.356270407 + 1.05 = -13.306270407 t = -13.306270407 Simplifying t = -13.306270407Solution
The solution to the problem is based on the solutions from the subproblems. t = {15.406270407, -13.306270407}
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