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Simplifying 2t(t + 1) = 7 Reorder the terms: 2t(1 + t) = 7 (1 * 2t + t * 2t) = 7 (2t + 2t2) = 7 Solving 2t + 2t2 = 7 Solving for variable 't'. Reorder the terms: -7 + 2t + 2t2 = 7 + -7 Combine like terms: 7 + -7 = 0 -7 + 2t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + t + t2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + t + 3.5 + t2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + t + t2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + t + t2 = 0 + 3.5 t + t2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 t + t2 = 3.5 The t term is t. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + t2 = 3.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + t2 = 3.5 + 0.25 Combine like terms: 3.5 + 0.25 = 3.75 1.25 + t2 = 3.75 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 3.75 Calculate the square root of the right side: 1.936491673 Break this problem into two subproblems by setting (t + 0.5) equal to 1.936491673 and -1.936491673.Subproblem 1
t + 0.5 = 1.936491673 Simplifying t + 0.5 = 1.936491673 Reorder the terms: 0.5 + t = 1.936491673 Solving 0.5 + t = 1.936491673 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = 1.936491673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 1.936491673 + -0.5 t = 1.936491673 + -0.5 Combine like terms: 1.936491673 + -0.5 = 1.436491673 t = 1.436491673 Simplifying t = 1.436491673Subproblem 2
t + 0.5 = -1.936491673 Simplifying t + 0.5 = -1.936491673 Reorder the terms: 0.5 + t = -1.936491673 Solving 0.5 + t = -1.936491673 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + t = -1.936491673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -1.936491673 + -0.5 t = -1.936491673 + -0.5 Combine like terms: -1.936491673 + -0.5 = -2.436491673 t = -2.436491673 Simplifying t = -2.436491673Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.436491673, -2.436491673}
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