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Simplifying (2t + -3) * 2 = 2t2 + 5t + -26 Reorder the terms: (-3 + 2t) * 2 = 2t2 + 5t + -26 Reorder the terms for easier multiplication: 2(-3 + 2t) = 2t2 + 5t + -26 (-3 * 2 + 2t * 2) = 2t2 + 5t + -26 (-6 + 4t) = 2t2 + 5t + -26 Reorder the terms: -6 + 4t = -26 + 5t + 2t2 Solving -6 + 4t = -26 + 5t + 2t2 Solving for variable 't'. Reorder the terms: -6 + 26 + 4t + -5t + -2t2 = -26 + 5t + 2t2 + 26 + -5t + -2t2 Combine like terms: -6 + 26 = 20 20 + 4t + -5t + -2t2 = -26 + 5t + 2t2 + 26 + -5t + -2t2 Combine like terms: 4t + -5t = -1t 20 + -1t + -2t2 = -26 + 5t + 2t2 + 26 + -5t + -2t2 Reorder the terms: 20 + -1t + -2t2 = -26 + 26 + 5t + -5t + 2t2 + -2t2 Combine like terms: -26 + 26 = 0 20 + -1t + -2t2 = 0 + 5t + -5t + 2t2 + -2t2 20 + -1t + -2t2 = 5t + -5t + 2t2 + -2t2 Combine like terms: 5t + -5t = 0 20 + -1t + -2t2 = 0 + 2t2 + -2t2 20 + -1t + -2t2 = 2t2 + -2t2 Combine like terms: 2t2 + -2t2 = 0 20 + -1t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -10 + 0.5t + t2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + 0.5t + 10 + t2 = 0 + 10 Reorder the terms: -10 + 10 + 0.5t + t2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + 0.5t + t2 = 0 + 10 0.5t + t2 = 0 + 10 Combine like terms: 0 + 10 = 10 0.5t + t2 = 10 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 = 10 + 0.0625 Reorder the terms: 0.0625 + 0.5t + t2 = 10 + 0.0625 Combine like terms: 10 + 0.0625 = 10.0625 0.0625 + 0.5t + t2 = 10.0625 Factor a perfect square on the left side: (t + 0.25)(t + 0.25) = 10.0625 Calculate the square root of the right side: 3.172144385 Break this problem into two subproblems by setting (t + 0.25) equal to 3.172144385 and -3.172144385.Subproblem 1
t + 0.25 = 3.172144385 Simplifying t + 0.25 = 3.172144385 Reorder the terms: 0.25 + t = 3.172144385 Solving 0.25 + t = 3.172144385 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 = 3.172144385 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + t = 3.172144385 + -0.25 t = 3.172144385 + -0.25 Combine like terms: 3.172144385 + -0.25 = 2.922144385 t = 2.922144385 Simplifying t = 2.922144385Subproblem 2
t + 0.25 = -3.172144385 Simplifying t + 0.25 = -3.172144385 Reorder the terms: 0.25 + t = -3.172144385 Solving 0.25 + t = -3.172144385 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 = -3.172144385 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + t = -3.172144385 + -0.25 t = -3.172144385 + -0.25 Combine like terms: -3.172144385 + -0.25 = -3.422144385 t = -3.422144385 Simplifying t = -3.422144385Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.922144385, -3.422144385}
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