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Simplifying 4t2 + 9t + 1 = 0 Reorder the terms: 1 + 9t + 4t2 = 0 Solving 1 + 9t + 4t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.25 + 2.25t + t2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + 2.25t + -0.25 + t2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + 2.25t + t2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + 2.25t + t2 = 0 + -0.25 2.25t + t2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 2.25t + t2 = -0.25 The t term is 2.25t. Take half its coefficient (1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. 2.25t + 1.265625 + t2 = -0.25 + 1.265625 Reorder the terms: 1.265625 + 2.25t + t2 = -0.25 + 1.265625 Combine like terms: -0.25 + 1.265625 = 1.015625 1.265625 + 2.25t + t2 = 1.015625 Factor a perfect square on the left side: (t + 1.125)(t + 1.125) = 1.015625 Calculate the square root of the right side: 1.007782219 Break this problem into two subproblems by setting (t + 1.125) equal to 1.007782219 and -1.007782219.Subproblem 1
t + 1.125 = 1.007782219 Simplifying t + 1.125 = 1.007782219 Reorder the terms: 1.125 + t = 1.007782219 Solving 1.125 + t = 1.007782219 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + t = 1.007782219 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + t = 1.007782219 + -1.125 t = 1.007782219 + -1.125 Combine like terms: 1.007782219 + -1.125 = -0.117217781 t = -0.117217781 Simplifying t = -0.117217781Subproblem 2
t + 1.125 = -1.007782219 Simplifying t + 1.125 = -1.007782219 Reorder the terms: 1.125 + t = -1.007782219 Solving 1.125 + t = -1.007782219 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + t = -1.007782219 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + t = -1.007782219 + -1.125 t = -1.007782219 + -1.125 Combine like terms: -1.007782219 + -1.125 = -2.132782219 t = -2.132782219 Simplifying t = -2.132782219Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.117217781, -2.132782219}
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