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Simplifying 2t * 2t + 5t + -3 = 0 Reorder the terms for easier multiplication: 2 * 2t * t + 5t + -3 = 0 Multiply 2 * 2 4t * t + 5t + -3 = 0 Multiply t * t 4t2 + 5t + -3 = 0 Reorder the terms: -3 + 5t + 4t2 = 0 Solving -3 + 5t + 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'. -0.75 + 1.25t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 1.25t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 1.25t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 1.25t + t2 = 0 + 0.75 1.25t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 1.25t + t2 = 0.75 The t term is 1.25t. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25t + 0.390625 + t2 = 0.75 + 0.390625 Reorder the terms: 0.390625 + 1.25t + t2 = 0.75 + 0.390625 Combine like terms: 0.75 + 0.390625 = 1.140625 0.390625 + 1.25t + t2 = 1.140625 Factor a perfect square on the left side: (t + 0.625)(t + 0.625) = 1.140625 Calculate the square root of the right side: 1.068000468 Break this problem into two subproblems by setting (t + 0.625) equal to 1.068000468 and -1.068000468.Subproblem 1
t + 0.625 = 1.068000468 Simplifying t + 0.625 = 1.068000468 Reorder the terms: 0.625 + t = 1.068000468 Solving 0.625 + t = 1.068000468 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + t = 1.068000468 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + t = 1.068000468 + -0.625 t = 1.068000468 + -0.625 Combine like terms: 1.068000468 + -0.625 = 0.443000468 t = 0.443000468 Simplifying t = 0.443000468Subproblem 2
t + 0.625 = -1.068000468 Simplifying t + 0.625 = -1.068000468 Reorder the terms: 0.625 + t = -1.068000468 Solving 0.625 + t = -1.068000468 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + t = -1.068000468 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + t = -1.068000468 + -0.625 t = -1.068000468 + -0.625 Combine like terms: -1.068000468 + -0.625 = -1.693000468 t = -1.693000468 Simplifying t = -1.693000468Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.443000468, -1.693000468}
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