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Simplifying 12.5 = t2 + 3t Reorder the terms: 12.5 = 3t + t2 Solving 12.5 = 3t + t2 Solving for variable 't'. Reorder the terms: 12.5 + -3t + -1t2 = 3t + -3t + t2 + -1t2 Combine like terms: 3t + -3t = 0 12.5 + -3t + -1t2 = 0 + t2 + -1t2 12.5 + -3t + -1t2 = t2 + -1t2 Combine like terms: t2 + -1t2 = 0 12.5 + -3t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -12.5 + 3t + t2 = 0 Move the constant term to the right: Add '12.5' to each side of the equation. -12.5 + 3t + 12.5 + t2 = 0 + 12.5 Reorder the terms: -12.5 + 12.5 + 3t + t2 = 0 + 12.5 Combine like terms: -12.5 + 12.5 = 0.0 0.0 + 3t + t2 = 0 + 12.5 3t + t2 = 0 + 12.5 Combine like terms: 0 + 12.5 = 12.5 3t + t2 = 12.5 The t term is 3t. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3t + 2.25 + t2 = 12.5 + 2.25 Reorder the terms: 2.25 + 3t + t2 = 12.5 + 2.25 Combine like terms: 12.5 + 2.25 = 14.75 2.25 + 3t + t2 = 14.75 Factor a perfect square on the left side: (t + 1.5)(t + 1.5) = 14.75 Calculate the square root of the right side: 3.840572874 Break this problem into two subproblems by setting (t + 1.5) equal to 3.840572874 and -3.840572874.Subproblem 1
t + 1.5 = 3.840572874 Simplifying t + 1.5 = 3.840572874 Reorder the terms: 1.5 + t = 3.840572874 Solving 1.5 + t = 3.840572874 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = 3.840572874 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = 3.840572874 + -1.5 t = 3.840572874 + -1.5 Combine like terms: 3.840572874 + -1.5 = 2.340572874 t = 2.340572874 Simplifying t = 2.340572874Subproblem 2
t + 1.5 = -3.840572874 Simplifying t + 1.5 = -3.840572874 Reorder the terms: 1.5 + t = -3.840572874 Solving 1.5 + t = -3.840572874 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + t = -3.840572874 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + t = -3.840572874 + -1.5 t = -3.840572874 + -1.5 Combine like terms: -3.840572874 + -1.5 = -5.340572874 t = -5.340572874 Simplifying t = -5.340572874Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.340572874, -5.340572874}
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