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Simplifying 24t2 + 3t + -4 = 0 Reorder the terms: -4 + 3t + 24t2 = 0 Solving -4 + 3t + 24t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -0.1666666667 + 0.125t + t2 = 0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.125t + 0.1666666667 + t2 = 0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + 0.125t + t2 = 0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + 0.125t + t2 = 0 + 0.1666666667 0.125t + t2 = 0 + 0.1666666667 Combine like terms: 0 + 0.1666666667 = 0.1666666667 0.125t + t2 = 0.1666666667 The t term is 0.125t. Take half its coefficient (0.0625). Square it (0.00390625) and add it to both sides. Add '0.00390625' to each side of the equation. 0.125t + 0.00390625 + t2 = 0.1666666667 + 0.00390625 Reorder the terms: 0.00390625 + 0.125t + t2 = 0.1666666667 + 0.00390625 Combine like terms: 0.1666666667 + 0.00390625 = 0.1705729167 0.00390625 + 0.125t + t2 = 0.1705729167 Factor a perfect square on the left side: (t + 0.0625)(t + 0.0625) = 0.1705729167 Calculate the square root of the right side: 0.413004742 Break this problem into two subproblems by setting (t + 0.0625) equal to 0.413004742 and -0.413004742.Subproblem 1
t + 0.0625 = 0.413004742 Simplifying t + 0.0625 = 0.413004742 Reorder the terms: 0.0625 + t = 0.413004742 Solving 0.0625 + t = 0.413004742 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + t = 0.413004742 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + t = 0.413004742 + -0.0625 t = 0.413004742 + -0.0625 Combine like terms: 0.413004742 + -0.0625 = 0.350504742 t = 0.350504742 Simplifying t = 0.350504742Subproblem 2
t + 0.0625 = -0.413004742 Simplifying t + 0.0625 = -0.413004742 Reorder the terms: 0.0625 + t = -0.413004742 Solving 0.0625 + t = -0.413004742 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + t = -0.413004742 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + t = -0.413004742 + -0.0625 t = -0.413004742 + -0.0625 Combine like terms: -0.413004742 + -0.0625 = -0.475504742 t = -0.475504742 Simplifying t = -0.475504742Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.350504742, -0.475504742}
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