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Simplifying 100 = 0.2t2 + 3t + 10.4 Reorder the terms: 100 = 10.4 + 3t + 0.2t2 Solving 100 = 10.4 + 3t + 0.2t2 Solving for variable 't'. Combine like terms: 100 + -10.4 = 89.6 89.6 + -3t + -0.2t2 = 10.4 + 3t + 0.2t2 + -10.4 + -3t + -0.2t2 Reorder the terms: 89.6 + -3t + -0.2t2 = 10.4 + -10.4 + 3t + -3t + 0.2t2 + -0.2t2 Combine like terms: 10.4 + -10.4 = 0.0 89.6 + -3t + -0.2t2 = 0.0 + 3t + -3t + 0.2t2 + -0.2t2 89.6 + -3t + -0.2t2 = 3t + -3t + 0.2t2 + -0.2t2 Combine like terms: 3t + -3t = 0 89.6 + -3t + -0.2t2 = 0 + 0.2t2 + -0.2t2 89.6 + -3t + -0.2t2 = 0.2t2 + -0.2t2 Combine like terms: 0.2t2 + -0.2t2 = 0.0 89.6 + -3t + -0.2t2 = 0.0 Begin completing the square. Divide all terms by -0.2 the coefficient of the squared term: Divide each side by '-0.2'. -448 + 15t + t2 = 0 Move the constant term to the right: Add '448' to each side of the equation. -448 + 15t + 448 + t2 = 0 + 448 Reorder the terms: -448 + 448 + 15t + t2 = 0 + 448 Combine like terms: -448 + 448 = 0 0 + 15t + t2 = 0 + 448 15t + t2 = 0 + 448 Combine like terms: 0 + 448 = 448 15t + t2 = 448 The t term is 15t. Take half its coefficient (7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. 15t + 56.25 + t2 = 448 + 56.25 Reorder the terms: 56.25 + 15t + t2 = 448 + 56.25 Combine like terms: 448 + 56.25 = 504.25 56.25 + 15t + t2 = 504.25 Factor a perfect square on the left side: (t + 7.5)(t + 7.5) = 504.25 Calculate the square root of the right side: 22.455511573 Break this problem into two subproblems by setting (t + 7.5) equal to 22.455511573 and -22.455511573.Subproblem 1
t + 7.5 = 22.455511573 Simplifying t + 7.5 = 22.455511573 Reorder the terms: 7.5 + t = 22.455511573 Solving 7.5 + t = 22.455511573 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + t = 22.455511573 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + t = 22.455511573 + -7.5 t = 22.455511573 + -7.5 Combine like terms: 22.455511573 + -7.5 = 14.955511573 t = 14.955511573 Simplifying t = 14.955511573Subproblem 2
t + 7.5 = -22.455511573 Simplifying t + 7.5 = -22.455511573 Reorder the terms: 7.5 + t = -22.455511573 Solving 7.5 + t = -22.455511573 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + t = -22.455511573 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + t = -22.455511573 + -7.5 t = -22.455511573 + -7.5 Combine like terms: -22.455511573 + -7.5 = -29.955511573 t = -29.955511573 Simplifying t = -29.955511573Solution
The solution to the problem is based on the solutions from the subproblems. t = {14.955511573, -29.955511573}
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