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Simplifying -16t2 + 24t = -260 Reorder the terms: 24t + -16t2 = -260 Solving 24t + -16t2 = -260 Solving for variable 't'. Reorder the terms: 260 + 24t + -16t2 = -260 + 260 Combine like terms: -260 + 260 = 0 260 + 24t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(65 + 6t + -4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(65 + 6t + -4t2)' equal to zero and attempt to solve: Simplifying 65 + 6t + -4t2 = 0 Solving 65 + 6t + -4t2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -16.25 + -1.5t + t2 = 0 Move the constant term to the right: Add '16.25' to each side of the equation. -16.25 + -1.5t + 16.25 + t2 = 0 + 16.25 Reorder the terms: -16.25 + 16.25 + -1.5t + t2 = 0 + 16.25 Combine like terms: -16.25 + 16.25 = 0.00 0.00 + -1.5t + t2 = 0 + 16.25 -1.5t + t2 = 0 + 16.25 Combine like terms: 0 + 16.25 = 16.25 -1.5t + t2 = 16.25 The t term is -1.5t. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5t + 0.5625 + t2 = 16.25 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 16.25 + 0.5625 Combine like terms: 16.25 + 0.5625 = 16.8125 0.5625 + -1.5t + t2 = 16.8125 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 16.8125 Calculate the square root of the right side: 4.100304867 Break this problem into two subproblems by setting (t + -0.75) equal to 4.100304867 and -4.100304867.Subproblem 1
t + -0.75 = 4.100304867 Simplifying t + -0.75 = 4.100304867 Reorder the terms: -0.75 + t = 4.100304867 Solving -0.75 + t = 4.100304867 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = 4.100304867 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 4.100304867 + 0.75 t = 4.100304867 + 0.75 Combine like terms: 4.100304867 + 0.75 = 4.850304867 t = 4.850304867 Simplifying t = 4.850304867Subproblem 2
t + -0.75 = -4.100304867 Simplifying t + -0.75 = -4.100304867 Reorder the terms: -0.75 + t = -4.100304867 Solving -0.75 + t = -4.100304867 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + t = -4.100304867 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -4.100304867 + 0.75 t = -4.100304867 + 0.75 Combine like terms: -4.100304867 + 0.75 = -3.350304867 t = -3.350304867 Simplifying t = -3.350304867Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.850304867, -3.350304867}Solution
t = {4.850304867, -3.350304867}
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