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Simplifying 0 = -3t2 + 24t + -19 Reorder the terms: 0 = -19 + 24t + -3t2 Solving 0 = -19 + 24t + -3t2 Solving for variable 't'. Combine like terms: 0 + 19 = 19 19 + -24t + 3t2 = -19 + 24t + -3t2 + 19 + -24t + 3t2 Reorder the terms: 19 + -24t + 3t2 = -19 + 19 + 24t + -24t + -3t2 + 3t2 Combine like terms: -19 + 19 = 0 19 + -24t + 3t2 = 0 + 24t + -24t + -3t2 + 3t2 19 + -24t + 3t2 = 24t + -24t + -3t2 + 3t2 Combine like terms: 24t + -24t = 0 19 + -24t + 3t2 = 0 + -3t2 + 3t2 19 + -24t + 3t2 = -3t2 + 3t2 Combine like terms: -3t2 + 3t2 = 0 19 + -24t + 3t2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 6.333333333 + -8t + t2 = 0 Move the constant term to the right: Add '-6.333333333' to each side of the equation. 6.333333333 + -8t + -6.333333333 + t2 = 0 + -6.333333333 Reorder the terms: 6.333333333 + -6.333333333 + -8t + t2 = 0 + -6.333333333 Combine like terms: 6.333333333 + -6.333333333 = 0.000000000 0.000000000 + -8t + t2 = 0 + -6.333333333 -8t + t2 = 0 + -6.333333333 Combine like terms: 0 + -6.333333333 = -6.333333333 -8t + t2 = -6.333333333 The t term is -8t. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8t + 16 + t2 = -6.333333333 + 16 Reorder the terms: 16 + -8t + t2 = -6.333333333 + 16 Combine like terms: -6.333333333 + 16 = 9.666666667 16 + -8t + t2 = 9.666666667 Factor a perfect square on the left side: (t + -4)(t + -4) = 9.666666667 Calculate the square root of the right side: 3.109126351 Break this problem into two subproblems by setting (t + -4) equal to 3.109126351 and -3.109126351.Subproblem 1
t + -4 = 3.109126351 Simplifying t + -4 = 3.109126351 Reorder the terms: -4 + t = 3.109126351 Solving -4 + t = 3.109126351 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = 3.109126351 + 4 Combine like terms: -4 + 4 = 0 0 + t = 3.109126351 + 4 t = 3.109126351 + 4 Combine like terms: 3.109126351 + 4 = 7.109126351 t = 7.109126351 Simplifying t = 7.109126351Subproblem 2
t + -4 = -3.109126351 Simplifying t + -4 = -3.109126351 Reorder the terms: -4 + t = -3.109126351 Solving -4 + t = -3.109126351 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + t = -3.109126351 + 4 Combine like terms: -4 + 4 = 0 0 + t = -3.109126351 + 4 t = -3.109126351 + 4 Combine like terms: -3.109126351 + 4 = 0.890873649 t = 0.890873649 Simplifying t = 0.890873649Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.109126351, 0.890873649}
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