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Simplifying 5(t2 + -3) + 2t = t(t + -1) Reorder the terms: 5(-3 + t2) + 2t = t(t + -1) (-3 * 5 + t2 * 5) + 2t = t(t + -1) (-15 + 5t2) + 2t = t(t + -1) Reorder the terms: -15 + 2t + 5t2 = t(t + -1) Reorder the terms: -15 + 2t + 5t2 = t(-1 + t) -15 + 2t + 5t2 = (-1 * t + t * t) -15 + 2t + 5t2 = (-1t + t2) Solving -15 + 2t + 5t2 = -1t + t2 Solving for variable 't'. Reorder the terms: -15 + 2t + t + 5t2 + -1t2 = -1t + t2 + t + -1t2 Combine like terms: 2t + t = 3t -15 + 3t + 5t2 + -1t2 = -1t + t2 + t + -1t2 Combine like terms: 5t2 + -1t2 = 4t2 -15 + 3t + 4t2 = -1t + t2 + t + -1t2 Reorder the terms: -15 + 3t + 4t2 = -1t + t + t2 + -1t2 Combine like terms: -1t + t = 0 -15 + 3t + 4t2 = 0 + t2 + -1t2 -15 + 3t + 4t2 = t2 + -1t2 Combine like terms: t2 + -1t2 = 0 -15 + 3t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -3.75 + 0.75t + t2 = 0 Move the constant term to the right: Add '3.75' to each side of the equation. -3.75 + 0.75t + 3.75 + t2 = 0 + 3.75 Reorder the terms: -3.75 + 3.75 + 0.75t + t2 = 0 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + 0.75t + t2 = 0 + 3.75 0.75t + t2 = 0 + 3.75 Combine like terms: 0 + 3.75 = 3.75 0.75t + t2 = 3.75 The t term is 0.75t. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75t + 0.140625 + t2 = 3.75 + 0.140625 Reorder the terms: 0.140625 + 0.75t + t2 = 3.75 + 0.140625 Combine like terms: 3.75 + 0.140625 = 3.890625 0.140625 + 0.75t + t2 = 3.890625 Factor a perfect square on the left side: (t + 0.375)(t + 0.375) = 3.890625 Calculate the square root of the right side: 1.97246673 Break this problem into two subproblems by setting (t + 0.375) equal to 1.97246673 and -1.97246673.Subproblem 1
t + 0.375 = 1.97246673 Simplifying t + 0.375 = 1.97246673 Reorder the terms: 0.375 + t = 1.97246673 Solving 0.375 + t = 1.97246673 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = 1.97246673 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = 1.97246673 + -0.375 t = 1.97246673 + -0.375 Combine like terms: 1.97246673 + -0.375 = 1.59746673 t = 1.59746673 Simplifying t = 1.59746673Subproblem 2
t + 0.375 = -1.97246673 Simplifying t + 0.375 = -1.97246673 Reorder the terms: 0.375 + t = -1.97246673 Solving 0.375 + t = -1.97246673 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + t = -1.97246673 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + t = -1.97246673 + -0.375 t = -1.97246673 + -0.375 Combine like terms: -1.97246673 + -0.375 = -2.34746673 t = -2.34746673 Simplifying t = -2.34746673Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.59746673, -2.34746673}
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