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Simplifying 400 = 0.1t2 + 3t Reorder the terms: 400 = 3t + 0.1t2 Solving 400 = 3t + 0.1t2 Solving for variable 't'. Reorder the terms: 400 + -3t + -0.1t2 = 3t + -3t + 0.1t2 + -0.1t2 Combine like terms: 3t + -3t = 0 400 + -3t + -0.1t2 = 0 + 0.1t2 + -0.1t2 400 + -3t + -0.1t2 = 0.1t2 + -0.1t2 Combine like terms: 0.1t2 + -0.1t2 = 0.0 400 + -3t + -0.1t2 = 0.0 Begin completing the square. Divide all terms by -0.1 the coefficient of the squared term: Divide each side by '-0.1'. -4000 + 30t + t2 = 0 Move the constant term to the right: Add '4000' to each side of the equation. -4000 + 30t + 4000 + t2 = 0 + 4000 Reorder the terms: -4000 + 4000 + 30t + t2 = 0 + 4000 Combine like terms: -4000 + 4000 = 0 0 + 30t + t2 = 0 + 4000 30t + t2 = 0 + 4000 Combine like terms: 0 + 4000 = 4000 30t + t2 = 4000 The t term is 30t. Take half its coefficient (15). Square it (225) and add it to both sides. Add '225' to each side of the equation. 30t + 225 + t2 = 4000 + 225 Reorder the terms: 225 + 30t + t2 = 4000 + 225 Combine like terms: 4000 + 225 = 4225 225 + 30t + t2 = 4225 Factor a perfect square on the left side: (t + 15)(t + 15) = 4225 Calculate the square root of the right side: 65 Break this problem into two subproblems by setting (t + 15) equal to 65 and -65.Subproblem 1
t + 15 = 65 Simplifying t + 15 = 65 Reorder the terms: 15 + t = 65 Solving 15 + t = 65 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + t = 65 + -15 Combine like terms: 15 + -15 = 0 0 + t = 65 + -15 t = 65 + -15 Combine like terms: 65 + -15 = 50 t = 50 Simplifying t = 50Subproblem 2
t + 15 = -65 Simplifying t + 15 = -65 Reorder the terms: 15 + t = -65 Solving 15 + t = -65 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + t = -65 + -15 Combine like terms: 15 + -15 = 0 0 + t = -65 + -15 t = -65 + -15 Combine like terms: -65 + -15 = -80 t = -80 Simplifying t = -80Solution
The solution to the problem is based on the solutions from the subproblems. t = {50, -80}
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