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Simplifying 200 = 6t2 + 4.5t + 74 Reorder the terms: 200 = 74 + 4.5t + 6t2 Solving 200 = 74 + 4.5t + 6t2 Solving for variable 't'. Combine like terms: 200 + -74 = 126 126 + -4.5t + -6t2 = 74 + 4.5t + 6t2 + -74 + -4.5t + -6t2 Reorder the terms: 126 + -4.5t + -6t2 = 74 + -74 + 4.5t + -4.5t + 6t2 + -6t2 Combine like terms: 74 + -74 = 0 126 + -4.5t + -6t2 = 0 + 4.5t + -4.5t + 6t2 + -6t2 126 + -4.5t + -6t2 = 4.5t + -4.5t + 6t2 + -6t2 Combine like terms: 4.5t + -4.5t = 0.0 126 + -4.5t + -6t2 = 0.0 + 6t2 + -6t2 126 + -4.5t + -6t2 = 6t2 + -6t2 Combine like terms: 6t2 + -6t2 = 0 126 + -4.5t + -6t2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -21 + 0.75t + t2 = 0 Move the constant term to the right: Add '21' to each side of the equation. -21 + 0.75t + 21 + t2 = 0 + 21 Reorder the terms: -21 + 21 + 0.75t + t2 = 0 + 21 Combine like terms: -21 + 21 = 0 0 + 0.75t + t2 = 0 + 21 0.75t + t2 = 0 + 21 Combine like terms: 0 + 21 = 21 0.75t + t2 = 21 The t term is 0.75t. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75t + 0.140625 + t2 = 21 + 0.140625 Reorder the terms: 0.140625 + 0.75t + t2 = 21 + 0.140625 Combine like terms: 21 + 0.140625 = 21.140625 0.140625 + 0.75t + t2 = 21.140625 Factor a perfect square on the left side: (t + 0.375)(t + 0.375) = 21.140625 Calculate the square root of the right side: 4.597893539 Break this problem into two subproblems by setting (t + 0.375) equal to 4.597893539 and -4.597893539.Subproblem 1
t + 0.375 = 4.597893539 Simplifying t + 0.375 = 4.597893539 Reorder the terms: 0.375 + t = 4.597893539 Solving 0.375 + t = 4.597893539 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = 4.597893539 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = 4.597893539 + -0.375 t = 4.597893539 + -0.375 Combine like terms: 4.597893539 + -0.375 = 4.222893539 t = 4.222893539 Simplifying t = 4.222893539Subproblem 2
t + 0.375 = -4.597893539 Simplifying t + 0.375 = -4.597893539 Reorder the terms: 0.375 + t = -4.597893539 Solving 0.375 + t = -4.597893539 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = -4.597893539 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = -4.597893539 + -0.375 t = -4.597893539 + -0.375 Combine like terms: -4.597893539 + -0.375 = -4.972893539 t = -4.972893539 Simplifying t = -4.972893539Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.222893539, -4.972893539}
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