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Simplifying 0 = 12 + 7.5t + -5t2 Solving 0 = 12 + 7.5t + -5t2 Solving for variable 't'. Combine like terms: 0 + -12 = -12 -12 + -7.5t + 5t2 = 12 + 7.5t + -5t2 + -12 + -7.5t + 5t2 Reorder the terms: -12 + -7.5t + 5t2 = 12 + -12 + 7.5t + -7.5t + -5t2 + 5t2 Combine like terms: 12 + -12 = 0 -12 + -7.5t + 5t2 = 0 + 7.5t + -7.5t + -5t2 + 5t2 -12 + -7.5t + 5t2 = 7.5t + -7.5t + -5t2 + 5t2 Combine like terms: 7.5t + -7.5t = 0.0 -12 + -7.5t + 5t2 = 0.0 + -5t2 + 5t2 -12 + -7.5t + 5t2 = -5t2 + 5t2 Combine like terms: -5t2 + 5t2 = 0 -12 + -7.5t + 5t2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2.4 + -1.5t + t2 = 0 Move the constant term to the right: Add '2.4' to each side of the equation. -2.4 + -1.5t + 2.4 + t2 = 0 + 2.4 Reorder the terms: -2.4 + 2.4 + -1.5t + t2 = 0 + 2.4 Combine like terms: -2.4 + 2.4 = 0.0 0.0 + -1.5t + t2 = 0 + 2.4 -1.5t + t2 = 0 + 2.4 Combine like terms: 0 + 2.4 = 2.4 -1.5t + t2 = 2.4 The t term is -1.5t. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5t + 0.5625 + t2 = 2.4 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 2.4 + 0.5625 Combine like terms: 2.4 + 0.5625 = 2.9625 0.5625 + -1.5t + t2 = 2.9625 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 2.9625 Calculate the square root of the right side: 1.721191448 Break this problem into two subproblems by setting (t + -0.75) equal to 1.721191448 and -1.721191448.Subproblem 1
t + -0.75 = 1.721191448 Simplifying t + -0.75 = 1.721191448 Reorder the terms: -0.75 + t = 1.721191448 Solving -0.75 + t = 1.721191448 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = 1.721191448 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 1.721191448 + 0.75 t = 1.721191448 + 0.75 Combine like terms: 1.721191448 + 0.75 = 2.471191448 t = 2.471191448 Simplifying t = 2.471191448Subproblem 2
t + -0.75 = -1.721191448 Simplifying t + -0.75 = -1.721191448 Reorder the terms: -0.75 + t = -1.721191448 Solving -0.75 + t = -1.721191448 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = -1.721191448 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -1.721191448 + 0.75 t = -1.721191448 + 0.75 Combine like terms: -1.721191448 + 0.75 = -0.971191448 t = -0.971191448 Simplifying t = -0.971191448Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.471191448, -0.971191448}
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