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Simplifying 5t2 + 24t + 24 = 0 Reorder the terms: 24 + 24t + 5t2 = 0 Solving 24 + 24t + 5t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 4.8 + 4.8t + t2 = 0 Move the constant term to the right: Add '-4.8' to each side of the equation. 4.8 + 4.8t + -4.8 + t2 = 0 + -4.8 Reorder the terms: 4.8 + -4.8 + 4.8t + t2 = 0 + -4.8 Combine like terms: 4.8 + -4.8 = 0.0 0.0 + 4.8t + t2 = 0 + -4.8 4.8t + t2 = 0 + -4.8 Combine like terms: 0 + -4.8 = -4.8 4.8t + t2 = -4.8 The t term is 4.8t. Take half its coefficient (2.4). Square it (5.76) and add it to both sides. Add '5.76' to each side of the equation. 4.8t + 5.76 + t2 = -4.8 + 5.76 Reorder the terms: 5.76 + 4.8t + t2 = -4.8 + 5.76 Combine like terms: -4.8 + 5.76 = 0.96 5.76 + 4.8t + t2 = 0.96 Factor a perfect square on the left side: (t + 2.4)(t + 2.4) = 0.96 Calculate the square root of the right side: 0.979795897 Break this problem into two subproblems by setting (t + 2.4) equal to 0.979795897 and -0.979795897.Subproblem 1
t + 2.4 = 0.979795897 Simplifying t + 2.4 = 0.979795897 Reorder the terms: 2.4 + t = 0.979795897 Solving 2.4 + t = 0.979795897 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.4' to each side of the equation. 2.4 + -2.4 + t = 0.979795897 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + t = 0.979795897 + -2.4 t = 0.979795897 + -2.4 Combine like terms: 0.979795897 + -2.4 = -1.420204103 t = -1.420204103 Simplifying t = -1.420204103Subproblem 2
t + 2.4 = -0.979795897 Simplifying t + 2.4 = -0.979795897 Reorder the terms: 2.4 + t = -0.979795897 Solving 2.4 + t = -0.979795897 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.4' to each side of the equation. 2.4 + -2.4 + t = -0.979795897 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + t = -0.979795897 + -2.4 t = -0.979795897 + -2.4 Combine like terms: -0.979795897 + -2.4 = -3.379795897 t = -3.379795897 Simplifying t = -3.379795897Solution
The solution to the problem is based on the solutions from the subproblems. t = {-1.420204103, -3.379795897}
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