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Simplifying -5t2 + 84t = 176.4 Reorder the terms: 84t + -5t2 = 176.4 Solving 84t + -5t2 = 176.4 Solving for variable 't'. Reorder the terms: -176.4 + 84t + -5t2 = 176.4 + -176.4 Combine like terms: 176.4 + -176.4 = 0.0 -176.4 + 84t + -5t2 = 0.0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 35.28 + -16.8t + t2 = 0.0 Move the constant term to the right: Add '-35.28' to each side of the equation. 35.28 + -16.8t + -35.28 + t2 = 0.0 + -35.28 Reorder the terms: 35.28 + -35.28 + -16.8t + t2 = 0.0 + -35.28 Combine like terms: 35.28 + -35.28 = 0.00 0.00 + -16.8t + t2 = 0.0 + -35.28 -16.8t + t2 = 0.0 + -35.28 Combine like terms: 0.0 + -35.28 = -35.28 -16.8t + t2 = -35.28 The t term is -16.8t. Take half its coefficient (-8.4). Square it (70.56) and add it to both sides. Add '70.56' to each side of the equation. -16.8t + 70.56 + t2 = -35.28 + 70.56 Reorder the terms: 70.56 + -16.8t + t2 = -35.28 + 70.56 Combine like terms: -35.28 + 70.56 = 35.28 70.56 + -16.8t + t2 = 35.28 Factor a perfect square on the left side: (t + -8.4)(t + -8.4) = 35.28 Calculate the square root of the right side: 5.939696962 Break this problem into two subproblems by setting (t + -8.4) equal to 5.939696962 and -5.939696962.Subproblem 1
t + -8.4 = 5.939696962 Simplifying t + -8.4 = 5.939696962 Reorder the terms: -8.4 + t = 5.939696962 Solving -8.4 + t = 5.939696962 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '8.4' to each side of the equation. -8.4 + 8.4 + t = 5.939696962 + 8.4 Combine like terms: -8.4 + 8.4 = 0.0 0.0 + t = 5.939696962 + 8.4 t = 5.939696962 + 8.4 Combine like terms: 5.939696962 + 8.4 = 14.339696962 t = 14.339696962 Simplifying t = 14.339696962Subproblem 2
t + -8.4 = -5.939696962 Simplifying t + -8.4 = -5.939696962 Reorder the terms: -8.4 + t = -5.939696962 Solving -8.4 + t = -5.939696962 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '8.4' to each side of the equation. -8.4 + 8.4 + t = -5.939696962 + 8.4 Combine like terms: -8.4 + 8.4 = 0.0 0.0 + t = -5.939696962 + 8.4 t = -5.939696962 + 8.4 Combine like terms: -5.939696962 + 8.4 = 2.460303038 t = 2.460303038 Simplifying t = 2.460303038Solution
The solution to the problem is based on the solutions from the subproblems. t = {14.339696962, 2.460303038}
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