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Simplifying 4t2 + (14t(t + 0.5)) * 0.5 = 24 Reorder the terms: 4t2 + (14t(0.5 + t)) * 0.5 = 24 4t2 + ((0.5 * 14t + t * 14t)) * 0.5 = 24 4t2 + ((7t + 14t2)) * 0.5 = 24 Reorder the terms for easier multiplication: 4t2 + 0.5(7t + 14t2) = 24 4t2 + (7t * 0.5 + 14t2 * 0.5) = 24 4t2 + (3.5t + 7t2) = 24 Reorder the terms: 3.5t + 4t2 + 7t2 = 24 Combine like terms: 4t2 + 7t2 = 11t2 3.5t + 11t2 = 24 Solving 3.5t + 11t2 = 24 Solving for variable 't'. Reorder the terms: -24 + 3.5t + 11t2 = 24 + -24 Combine like terms: 24 + -24 = 0 -24 + 3.5t + 11t2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -2.181818182 + 0.3181818182t + t2 = 0 Move the constant term to the right: Add '2.181818182' to each side of the equation. -2.181818182 + 0.3181818182t + 2.181818182 + t2 = 0 + 2.181818182 Reorder the terms: -2.181818182 + 2.181818182 + 0.3181818182t + t2 = 0 + 2.181818182 Combine like terms: -2.181818182 + 2.181818182 = 0.000000000 0.000000000 + 0.3181818182t + t2 = 0 + 2.181818182 0.3181818182t + t2 = 0 + 2.181818182 Combine like terms: 0 + 2.181818182 = 2.181818182 0.3181818182t + t2 = 2.181818182 The t term is 0.3181818182t. Take half its coefficient (0.1590909091). Square it (0.02530991736) and add it to both sides. Add '0.02530991736' to each side of the equation. 0.3181818182t + 0.02530991736 + t2 = 2.181818182 + 0.02530991736 Reorder the terms: 0.02530991736 + 0.3181818182t + t2 = 2.181818182 + 0.02530991736 Combine like terms: 2.181818182 + 0.02530991736 = 2.20712809936 0.02530991736 + 0.3181818182t + t2 = 2.20712809936 Factor a perfect square on the left side: (t + 0.1590909091)(t + 0.1590909091) = 2.20712809936 Calculate the square root of the right side: 1.485640636 Break this problem into two subproblems by setting (t + 0.1590909091) equal to 1.485640636 and -1.485640636.Subproblem 1
t + 0.1590909091 = 1.485640636 Simplifying t + 0.1590909091 = 1.485640636 Reorder the terms: 0.1590909091 + t = 1.485640636 Solving 0.1590909091 + t = 1.485640636 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1590909091' to each side of the equation. 0.1590909091 + -0.1590909091 + t = 1.485640636 + -0.1590909091 Combine like terms: 0.1590909091 + -0.1590909091 = 0.0000000000 0.0000000000 + t = 1.485640636 + -0.1590909091 t = 1.485640636 + -0.1590909091 Combine like terms: 1.485640636 + -0.1590909091 = 1.3265497269 t = 1.3265497269 Simplifying t = 1.3265497269Subproblem 2
t + 0.1590909091 = -1.485640636 Simplifying t + 0.1590909091 = -1.485640636 Reorder the terms: 0.1590909091 + t = -1.485640636 Solving 0.1590909091 + t = -1.485640636 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1590909091' to each side of the equation. 0.1590909091 + -0.1590909091 + t = -1.485640636 + -0.1590909091 Combine like terms: 0.1590909091 + -0.1590909091 = 0.0000000000 0.0000000000 + t = -1.485640636 + -0.1590909091 t = -1.485640636 + -0.1590909091 Combine like terms: -1.485640636 + -0.1590909091 = -1.6447315451 t = -1.6447315451 Simplifying t = -1.6447315451Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.3265497269, -1.6447315451}
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