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Simplifying 4t2 + 24t + 12 = 5 Reorder the terms: 12 + 24t + 4t2 = 5 Solving 12 + 24t + 4t2 = 5 Solving for variable 't'. Reorder the terms: 12 + -5 + 24t + 4t2 = 5 + -5 Combine like terms: 12 + -5 = 7 7 + 24t + 4t2 = 5 + -5 Combine like terms: 5 + -5 = 0 7 + 24t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1.75 + 6t + t2 = 0 Move the constant term to the right: Add '-1.75' to each side of the equation. 1.75 + 6t + -1.75 + t2 = 0 + -1.75 Reorder the terms: 1.75 + -1.75 + 6t + t2 = 0 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + 6t + t2 = 0 + -1.75 6t + t2 = 0 + -1.75 Combine like terms: 0 + -1.75 = -1.75 6t + t2 = -1.75 The t term is 6t. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6t + 9 + t2 = -1.75 + 9 Reorder the terms: 9 + 6t + t2 = -1.75 + 9 Combine like terms: -1.75 + 9 = 7.25 9 + 6t + t2 = 7.25 Factor a perfect square on the left side: (t + 3)(t + 3) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (t + 3) equal to 2.692582404 and -2.692582404.Subproblem 1
t + 3 = 2.692582404 Simplifying t + 3 = 2.692582404 Reorder the terms: 3 + t = 2.692582404 Solving 3 + t = 2.692582404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + t = 2.692582404 + -3 Combine like terms: 3 + -3 = 0 0 + t = 2.692582404 + -3 t = 2.692582404 + -3 Combine like terms: 2.692582404 + -3 = -0.307417596 t = -0.307417596 Simplifying t = -0.307417596Subproblem 2
t + 3 = -2.692582404 Simplifying t + 3 = -2.692582404 Reorder the terms: 3 + t = -2.692582404 Solving 3 + t = -2.692582404 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + t = -2.692582404 + -3 Combine like terms: 3 + -3 = 0 0 + t = -2.692582404 + -3 t = -2.692582404 + -3 Combine like terms: -2.692582404 + -3 = -5.692582404 t = -5.692582404 Simplifying t = -5.692582404Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.307417596, -5.692582404}
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