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Simplifying 12 = t2 + -1t + -12 Reorder the terms: 12 = -12 + -1t + t2 Solving 12 = -12 + -1t + t2 Solving for variable 't'. Combine like terms: 12 + 12 = 24 24 + t + -1t2 = -12 + -1t + t2 + 12 + t + -1t2 Reorder the terms: 24 + t + -1t2 = -12 + 12 + -1t + t + t2 + -1t2 Combine like terms: -12 + 12 = 0 24 + t + -1t2 = 0 + -1t + t + t2 + -1t2 24 + t + -1t2 = -1t + t + t2 + -1t2 Combine like terms: -1t + t = 0 24 + t + -1t2 = 0 + t2 + -1t2 24 + t + -1t2 = t2 + -1t2 Combine like terms: t2 + -1t2 = 0 24 + t + -1t2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -24 + -1t + t2 = 0 Move the constant term to the right: Add '24' to each side of the equation. -24 + -1t + 24 + t2 = 0 + 24 Reorder the terms: -24 + 24 + -1t + t2 = 0 + 24 Combine like terms: -24 + 24 = 0 0 + -1t + t2 = 0 + 24 -1t + t2 = 0 + 24 Combine like terms: 0 + 24 = 24 -1t + t2 = 24 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. -1t + 0.25 + t2 = 24 + 0.25 Reorder the terms: 0.25 + -1t + t2 = 24 + 0.25 Combine like terms: 24 + 0.25 = 24.25 0.25 + -1t + t2 = 24.25 Factor a perfect square on the left side: (t + 0.5)(t + 0.5) = 24.25 Calculate the square root of the right side: 4.924428901 Break this problem into two subproblems by setting (t + 0.5) equal to 4.924428901 and -4.924428901.Subproblem 1
t + 0.5 = 4.924428901 Simplifying t + 0.5 = 4.924428901 Reorder the terms: 0.5 + t = 4.924428901 Solving 0.5 + t = 4.924428901 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 = 4.924428901 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 4.924428901 + -0.5 t = 4.924428901 + -0.5 Combine like terms: 4.924428901 + -0.5 = 4.424428901 t = 4.424428901 Simplifying t = 4.424428901Subproblem 2
t + 0.5 = -4.924428901 Simplifying t + 0.5 = -4.924428901 Reorder the terms: 0.5 + t = -4.924428901 Solving 0.5 + t = -4.924428901 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 = -4.924428901 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -4.924428901 + -0.5 t = -4.924428901 + -0.5 Combine like terms: -4.924428901 + -0.5 = -5.424428901 t = -5.424428901 Simplifying t = -5.424428901Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.424428901, -5.424428901}
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