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Simplifying 16t(t + -5) + 5t = 7(3t + 3) + -9 Reorder the terms: 16t(-5 + t) + 5t = 7(3t + 3) + -9 (-5 * 16t + t * 16t) + 5t = 7(3t + 3) + -9 (-80t + 16t2) + 5t = 7(3t + 3) + -9 Reorder the terms: -80t + 5t + 16t2 = 7(3t + 3) + -9 Combine like terms: -80t + 5t = -75t -75t + 16t2 = 7(3t + 3) + -9 Reorder the terms: -75t + 16t2 = 7(3 + 3t) + -9 -75t + 16t2 = (3 * 7 + 3t * 7) + -9 -75t + 16t2 = (21 + 21t) + -9 Reorder the terms: -75t + 16t2 = 21 + -9 + 21t Combine like terms: 21 + -9 = 12 -75t + 16t2 = 12 + 21t Solving -75t + 16t2 = 12 + 21t Solving for variable 't'. Reorder the terms: -12 + -75t + -21t + 16t2 = 12 + 21t + -12 + -21t Combine like terms: -75t + -21t = -96t -12 + -96t + 16t2 = 12 + 21t + -12 + -21t Reorder the terms: -12 + -96t + 16t2 = 12 + -12 + 21t + -21t Combine like terms: 12 + -12 = 0 -12 + -96t + 16t2 = 0 + 21t + -21t -12 + -96t + 16t2 = 21t + -21t Combine like terms: 21t + -21t = 0 -12 + -96t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-3 + -24t + 4t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-3 + -24t + 4t2)' equal to zero and attempt to solve: Simplifying -3 + -24t + 4t2 = 0 Solving -3 + -24t + 4t2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + -6t + t2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -6t + 0.75 + t2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -6t + t2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -6t + t2 = 0 + 0.75 -6t + t2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -6t + t2 = 0.75 The t term is -6t. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6t + 9 + t2 = 0.75 + 9 Reorder the terms: 9 + -6t + t2 = 0.75 + 9 Combine like terms: 0.75 + 9 = 9.75 9 + -6t + t2 = 9.75 Factor a perfect square on the left side: (t + -3)(t + -3) = 9.75 Calculate the square root of the right side: 3.122498999 Break this problem into two subproblems by setting (t + -3) equal to 3.122498999 and -3.122498999.Subproblem 1
t + -3 = 3.122498999 Simplifying t + -3 = 3.122498999 Reorder the terms: -3 + t = 3.122498999 Solving -3 + t = 3.122498999 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = 3.122498999 + 3 Combine like terms: -3 + 3 = 0 0 + t = 3.122498999 + 3 t = 3.122498999 + 3 Combine like terms: 3.122498999 + 3 = 6.122498999 t = 6.122498999 Simplifying t = 6.122498999Subproblem 2
t + -3 = -3.122498999 Simplifying t + -3 = -3.122498999 Reorder the terms: -3 + t = -3.122498999 Solving -3 + t = -3.122498999 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + t = -3.122498999 + 3 Combine like terms: -3 + 3 = 0 0 + t = -3.122498999 + 3 t = -3.122498999 + 3 Combine like terms: -3.122498999 + 3 = -0.122498999 t = -0.122498999 Simplifying t = -0.122498999Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.122498999, -0.122498999}Solution
t = {6.122498999, -0.122498999}
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