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Simplifying 2t2 + 3t + 32 = 4600 Reorder the terms: 32 + 3t + 2t2 = 4600 Solving 32 + 3t + 2t2 = 4600 Solving for variable 't'. Reorder the terms: 32 + -4600 + 3t + 2t2 = 4600 + -4600 Combine like terms: 32 + -4600 = -4568 -4568 + 3t + 2t2 = 4600 + -4600 Combine like terms: 4600 + -4600 = 0 -4568 + 3t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2284 + 1.5t + t2 = 0 Move the constant term to the right: Add '2284' to each side of the equation. -2284 + 1.5t + 2284 + t2 = 0 + 2284 Reorder the terms: -2284 + 2284 + 1.5t + t2 = 0 + 2284 Combine like terms: -2284 + 2284 = 0 0 + 1.5t + t2 = 0 + 2284 1.5t + t2 = 0 + 2284 Combine like terms: 0 + 2284 = 2284 1.5t + t2 = 2284 The t term is 1.5t. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5t + 0.5625 + t2 = 2284 + 0.5625 Reorder the terms: 0.5625 + 1.5t + t2 = 2284 + 0.5625 Combine like terms: 2284 + 0.5625 = 2284.5625 0.5625 + 1.5t + t2 = 2284.5625 Factor a perfect square on the left side: (t + 0.75)(t + 0.75) = 2284.5625 Calculate the square root of the right side: 47.797097192 Break this problem into two subproblems by setting (t + 0.75) equal to 47.797097192 and -47.797097192.Subproblem 1
t + 0.75 = 47.797097192 Simplifying t + 0.75 = 47.797097192 Reorder the terms: 0.75 + t = 47.797097192 Solving 0.75 + t = 47.797097192 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + t = 47.797097192 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = 47.797097192 + -0.75 t = 47.797097192 + -0.75 Combine like terms: 47.797097192 + -0.75 = 47.047097192 t = 47.047097192 Simplifying t = 47.047097192Subproblem 2
t + 0.75 = -47.797097192 Simplifying t + 0.75 = -47.797097192 Reorder the terms: 0.75 + t = -47.797097192 Solving 0.75 + t = -47.797097192 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + t = -47.797097192 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + t = -47.797097192 + -0.75 t = -47.797097192 + -0.75 Combine like terms: -47.797097192 + -0.75 = -48.547097192 t = -48.547097192 Simplifying t = -48.547097192Solution
The solution to the problem is based on the solutions from the subproblems. t = {47.047097192, -48.547097192}
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