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Simplifying 0 = -16t2 + -6t + 24 Reorder the terms: 0 = 24 + -6t + -16t2 Solving 0 = 24 + -6t + -16t2 Solving for variable 't'. Combine like terms: 0 + -24 = -24 -24 + 6t + 16t2 = 24 + -6t + -16t2 + -24 + 6t + 16t2 Reorder the terms: -24 + 6t + 16t2 = 24 + -24 + -6t + 6t + -16t2 + 16t2 Combine like terms: 24 + -24 = 0 -24 + 6t + 16t2 = 0 + -6t + 6t + -16t2 + 16t2 -24 + 6t + 16t2 = -6t + 6t + -16t2 + 16t2 Combine like terms: -6t + 6t = 0 -24 + 6t + 16t2 = 0 + -16t2 + 16t2 -24 + 6t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -24 + 6t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-12 + 3t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-12 + 3t + 8t2)' equal to zero and attempt to solve: Simplifying -12 + 3t + 8t2 = 0 Solving -12 + 3t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -1.5 + 0.375t + t2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 0.375t + 1.5 + t2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 0.375t + t2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 0.375t + t2 = 0 + 1.5 0.375t + t2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 0.375t + t2 = 1.5 The t term is 0.375t. Take half its coefficient (0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. 0.375t + 0.03515625 + t2 = 1.5 + 0.03515625 Reorder the terms: 0.03515625 + 0.375t + t2 = 1.5 + 0.03515625 Combine like terms: 1.5 + 0.03515625 = 1.53515625 0.03515625 + 0.375t + t2 = 1.53515625 Factor a perfect square on the left side: (t + 0.1875)(t + 0.1875) = 1.53515625 Calculate the square root of the right side: 1.239014225 Break this problem into two subproblems by setting (t + 0.1875) equal to 1.239014225 and -1.239014225.Subproblem 1
t + 0.1875 = 1.239014225 Simplifying t + 0.1875 = 1.239014225 Reorder the terms: 0.1875 + t = 1.239014225 Solving 0.1875 + t = 1.239014225 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + t = 1.239014225 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + t = 1.239014225 + -0.1875 t = 1.239014225 + -0.1875 Combine like terms: 1.239014225 + -0.1875 = 1.051514225 t = 1.051514225 Simplifying t = 1.051514225Subproblem 2
t + 0.1875 = -1.239014225 Simplifying t + 0.1875 = -1.239014225 Reorder the terms: 0.1875 + t = -1.239014225 Solving 0.1875 + t = -1.239014225 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + t = -1.239014225 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + t = -1.239014225 + -0.1875 t = -1.239014225 + -0.1875 Combine like terms: -1.239014225 + -0.1875 = -1.426514225 t = -1.426514225 Simplifying t = -1.426514225Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.051514225, -1.426514225}Solution
t = {1.051514225, -1.426514225}
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