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Simplifying -12t2 + 120t + 1000 = 0 Reorder the terms: 1000 + 120t + -12t2 = 0 Solving 1000 + 120t + -12t2 = 0 Solving for variable 't'. Factor out the Greatest Common Factor (GCF), '4'. 4(250 + 30t + -3t2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(250 + 30t + -3t2)' equal to zero and attempt to solve: Simplifying 250 + 30t + -3t2 = 0 Solving 250 + 30t + -3t2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -83.33333333 + -10t + t2 = 0 Move the constant term to the right: Add '83.33333333' to each side of the equation. -83.33333333 + -10t + 83.33333333 + t2 = 0 + 83.33333333 Reorder the terms: -83.33333333 + 83.33333333 + -10t + t2 = 0 + 83.33333333 Combine like terms: -83.33333333 + 83.33333333 = 0.00000000 0.00000000 + -10t + t2 = 0 + 83.33333333 -10t + t2 = 0 + 83.33333333 Combine like terms: 0 + 83.33333333 = 83.33333333 -10t + t2 = 83.33333333 The t term is -10t. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10t + 25 + t2 = 83.33333333 + 25 Reorder the terms: 25 + -10t + t2 = 83.33333333 + 25 Combine like terms: 83.33333333 + 25 = 108.33333333 25 + -10t + t2 = 108.33333333 Factor a perfect square on the left side: (t + -5)(t + -5) = 108.33333333 Calculate the square root of the right side: 10.408329997 Break this problem into two subproblems by setting (t + -5) equal to 10.408329997 and -10.408329997.Subproblem 1
t + -5 = 10.408329997 Simplifying t + -5 = 10.408329997 Reorder the terms: -5 + t = 10.408329997 Solving -5 + t = 10.408329997 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + t = 10.408329997 + 5 Combine like terms: -5 + 5 = 0 0 + t = 10.408329997 + 5 t = 10.408329997 + 5 Combine like terms: 10.408329997 + 5 = 15.408329997 t = 15.408329997 Simplifying t = 15.408329997Subproblem 2
t + -5 = -10.408329997 Simplifying t + -5 = -10.408329997 Reorder the terms: -5 + t = -10.408329997 Solving -5 + t = -10.408329997 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + t = -10.408329997 + 5 Combine like terms: -5 + 5 = 0 0 + t = -10.408329997 + 5 t = -10.408329997 + 5 Combine like terms: -10.408329997 + 5 = -5.408329997 t = -5.408329997 Simplifying t = -5.408329997Solution
The solution to the problem is based on the solutions from the subproblems. t = {15.408329997, -5.408329997}Solution
t = {15.408329997, -5.408329997}
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