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Simplifying 4t2 = 9 + 2t Solving 4t2 = 9 + 2t Solving for variable 't'. Reorder the terms: -9 + -2t + 4t2 = 9 + 2t + -9 + -2t Reorder the terms: -9 + -2t + 4t2 = 9 + -9 + 2t + -2t Combine like terms: 9 + -9 = 0 -9 + -2t + 4t2 = 0 + 2t + -2t -9 + -2t + 4t2 = 2t + -2t Combine like terms: 2t + -2t = 0 -9 + -2t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2.25 + -0.5t + t2 = 0 Move the constant term to the right: Add '2.25' to each side of the equation. -2.25 + -0.5t + 2.25 + t2 = 0 + 2.25 Reorder the terms: -2.25 + 2.25 + -0.5t + t2 = 0 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + -0.5t + t2 = 0 + 2.25 -0.5t + t2 = 0 + 2.25 Combine like terms: 0 + 2.25 = 2.25 -0.5t + t2 = 2.25 The t term is -0.5t. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5t + 0.0625 + t2 = 2.25 + 0.0625 Reorder the terms: 0.0625 + -0.5t + t2 = 2.25 + 0.0625 Combine like terms: 2.25 + 0.0625 = 2.3125 0.0625 + -0.5t + t2 = 2.3125 Factor a perfect square on the left side: (t + -0.25)(t + -0.25) = 2.3125 Calculate the square root of the right side: 1.520690633 Break this problem into two subproblems by setting (t + -0.25) equal to 1.520690633 and -1.520690633.Subproblem 1
t + -0.25 = 1.520690633 Simplifying t + -0.25 = 1.520690633 Reorder the terms: -0.25 + t = 1.520690633 Solving -0.25 + t = 1.520690633 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + t = 1.520690633 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + t = 1.520690633 + 0.25 t = 1.520690633 + 0.25 Combine like terms: 1.520690633 + 0.25 = 1.770690633 t = 1.770690633 Simplifying t = 1.770690633Subproblem 2
t + -0.25 = -1.520690633 Simplifying t + -0.25 = -1.520690633 Reorder the terms: -0.25 + t = -1.520690633 Solving -0.25 + t = -1.520690633 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + t = -1.520690633 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + t = -1.520690633 + 0.25 t = -1.520690633 + 0.25 Combine like terms: -1.520690633 + 0.25 = -1.270690633 t = -1.270690633 Simplifying t = -1.270690633Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.770690633, -1.270690633}
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