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Simplifying t(2t + -3) = 40 Reorder the terms: t(-3 + 2t) = 40 (-3 * t + 2t * t) = 40 (-3t + 2t2) = 40 Solving -3t + 2t2 = 40 Solving for variable 't'. Reorder the terms: -40 + -3t + 2t2 = 40 + -40 Combine like terms: 40 + -40 = 0 -40 + -3t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -20 + -1.5t + t2 = 0 Move the constant term to the right: Add '20' to each side of the equation. -20 + -1.5t + 20 + t2 = 0 + 20 Reorder the terms: -20 + 20 + -1.5t + t2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + -1.5t + t2 = 0 + 20 -1.5t + t2 = 0 + 20 Combine like terms: 0 + 20 = 20 -1.5t + t2 = 20 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 = 20 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 20 + 0.5625 Combine like terms: 20 + 0.5625 = 20.5625 0.5625 + -1.5t + t2 = 20.5625 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 20.5625 Calculate the square root of the right side: 4.534589287 Break this problem into two subproblems by setting (t + -0.75) equal to 4.534589287 and -4.534589287.Subproblem 1
t + -0.75 = 4.534589287 Simplifying t + -0.75 = 4.534589287 Reorder the terms: -0.75 + t = 4.534589287 Solving -0.75 + t = 4.534589287 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 = 4.534589287 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 4.534589287 + 0.75 t = 4.534589287 + 0.75 Combine like terms: 4.534589287 + 0.75 = 5.284589287 t = 5.284589287 Simplifying t = 5.284589287Subproblem 2
t + -0.75 = -4.534589287 Simplifying t + -0.75 = -4.534589287 Reorder the terms: -0.75 + t = -4.534589287 Solving -0.75 + t = -4.534589287 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 = -4.534589287 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -4.534589287 + 0.75 t = -4.534589287 + 0.75 Combine like terms: -4.534589287 + 0.75 = -3.784589287 t = -3.784589287 Simplifying t = -3.784589287Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.284589287, -3.784589287}
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