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Simplifying 47t(t + -6) = 180 Reorder the terms: 47t(-6 + t) = 180 (-6 * 47t + t * 47t) = 180 (-282t + 47t2) = 180 Solving -282t + 47t2 = 180 Solving for variable 't'. Reorder the terms: -180 + -282t + 47t2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + -282t + 47t2 = 0 Begin completing the square. Divide all terms by 47 the coefficient of the squared term: Divide each side by '47'. -3.829787234 + -6t + t2 = 0 Move the constant term to the right: Add '3.829787234' to each side of the equation. -3.829787234 + -6t + 3.829787234 + t2 = 0 + 3.829787234 Reorder the terms: -3.829787234 + 3.829787234 + -6t + t2 = 0 + 3.829787234 Combine like terms: -3.829787234 + 3.829787234 = 0.000000000 0.000000000 + -6t + t2 = 0 + 3.829787234 -6t + t2 = 0 + 3.829787234 Combine like terms: 0 + 3.829787234 = 3.829787234 -6t + t2 = 3.829787234 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 = 3.829787234 + 9 Reorder the terms: 9 + -6t + t2 = 3.829787234 + 9 Combine like terms: 3.829787234 + 9 = 12.829787234 9 + -6t + t2 = 12.829787234 Factor a perfect square on the left side: (t + -3)(t + -3) = 12.829787234 Calculate the square root of the right side: 3.581869237 Break this problem into two subproblems by setting (t + -3) equal to 3.581869237 and -3.581869237.Subproblem 1
t + -3 = 3.581869237 Simplifying t + -3 = 3.581869237 Reorder the terms: -3 + t = 3.581869237 Solving -3 + t = 3.581869237 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 = 3.581869237 + 3 Combine like terms: -3 + 3 = 0 0 + t = 3.581869237 + 3 t = 3.581869237 + 3 Combine like terms: 3.581869237 + 3 = 6.581869237 t = 6.581869237 Simplifying t = 6.581869237Subproblem 2
t + -3 = -3.581869237 Simplifying t + -3 = -3.581869237 Reorder the terms: -3 + t = -3.581869237 Solving -3 + t = -3.581869237 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 = -3.581869237 + 3 Combine like terms: -3 + 3 = 0 0 + t = -3.581869237 + 3 t = -3.581869237 + 3 Combine like terms: -3.581869237 + 3 = -0.581869237 t = -0.581869237 Simplifying t = -0.581869237Solution
The solution to the problem is based on the solutions from the subproblems. t = {6.581869237, -0.581869237}
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