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Simplifying 6.9t + 15 = -4.9t2 + 27t Reorder the terms: 15 + 6.9t = -4.9t2 + 27t Reorder the terms: 15 + 6.9t = 27t + -4.9t2 Solving 15 + 6.9t = 27t + -4.9t2 Solving for variable 't'. Combine like terms: 6.9t + -27t = -20.1t 15 + -20.1t + 4.9t2 = 27t + -4.9t2 + -27t + 4.9t2 Reorder the terms: 15 + -20.1t + 4.9t2 = 27t + -27t + -4.9t2 + 4.9t2 Combine like terms: 27t + -27t = 0 15 + -20.1t + 4.9t2 = 0 + -4.9t2 + 4.9t2 15 + -20.1t + 4.9t2 = -4.9t2 + 4.9t2 Combine like terms: -4.9t2 + 4.9t2 = 0.0 15 + -20.1t + 4.9t2 = 0.0 Begin completing the square. Divide all terms by 4.9 the coefficient of the squared term: Divide each side by '4.9'. 3.06122449 + -4.102040816t + t2 = 0 Move the constant term to the right: Add '-3.06122449' to each side of the equation. 3.06122449 + -4.102040816t + -3.06122449 + t2 = 0 + -3.06122449 Reorder the terms: 3.06122449 + -3.06122449 + -4.102040816t + t2 = 0 + -3.06122449 Combine like terms: 3.06122449 + -3.06122449 = 0.00000000 0.00000000 + -4.102040816t + t2 = 0 + -3.06122449 -4.102040816t + t2 = 0 + -3.06122449 Combine like terms: 0 + -3.06122449 = -3.06122449 -4.102040816t + t2 = -3.06122449 The t term is -4.102040816t. Take half its coefficient (-2.051020408). Square it (4.206684714) and add it to both sides. Add '4.206684714' to each side of the equation. -4.102040816t + 4.206684714 + t2 = -3.06122449 + 4.206684714 Reorder the terms: 4.206684714 + -4.102040816t + t2 = -3.06122449 + 4.206684714 Combine like terms: -3.06122449 + 4.206684714 = 1.145460224 4.206684714 + -4.102040816t + t2 = 1.145460224 Factor a perfect square on the left side: (t + -2.051020408)(t + -2.051020408) = 1.145460224 Calculate the square root of the right side: 1.070261755 Break this problem into two subproblems by setting (t + -2.051020408) equal to 1.070261755 and -1.070261755.Subproblem 1
t + -2.051020408 = 1.070261755 Simplifying t + -2.051020408 = 1.070261755 Reorder the terms: -2.051020408 + t = 1.070261755 Solving -2.051020408 + t = 1.070261755 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.051020408' to each side of the equation. -2.051020408 + 2.051020408 + t = 1.070261755 + 2.051020408 Combine like terms: -2.051020408 + 2.051020408 = 0.000000000 0.000000000 + t = 1.070261755 + 2.051020408 t = 1.070261755 + 2.051020408 Combine like terms: 1.070261755 + 2.051020408 = 3.121282163 t = 3.121282163 Simplifying t = 3.121282163Subproblem 2
t + -2.051020408 = -1.070261755 Simplifying t + -2.051020408 = -1.070261755 Reorder the terms: -2.051020408 + t = -1.070261755 Solving -2.051020408 + t = -1.070261755 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.051020408' to each side of the equation. -2.051020408 + 2.051020408 + t = -1.070261755 + 2.051020408 Combine like terms: -2.051020408 + 2.051020408 = 0.000000000 0.000000000 + t = -1.070261755 + 2.051020408 t = -1.070261755 + 2.051020408 Combine like terms: -1.070261755 + 2.051020408 = 0.980758653 t = 0.980758653 Simplifying t = 0.980758653Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.121282163, 0.980758653}
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