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Simplifying 0 = -2t2 + 87t + 90 Reorder the terms: 0 = 90 + 87t + -2t2 Solving 0 = 90 + 87t + -2t2 Solving for variable 't'. Combine like terms: 0 + -90 = -90 -90 + -87t + 2t2 = 90 + 87t + -2t2 + -90 + -87t + 2t2 Reorder the terms: -90 + -87t + 2t2 = 90 + -90 + 87t + -87t + -2t2 + 2t2 Combine like terms: 90 + -90 = 0 -90 + -87t + 2t2 = 0 + 87t + -87t + -2t2 + 2t2 -90 + -87t + 2t2 = 87t + -87t + -2t2 + 2t2 Combine like terms: 87t + -87t = 0 -90 + -87t + 2t2 = 0 + -2t2 + 2t2 -90 + -87t + 2t2 = -2t2 + 2t2 Combine like terms: -2t2 + 2t2 = 0 -90 + -87t + 2t2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -45 + -43.5t + t2 = 0 Move the constant term to the right: Add '45' to each side of the equation. -45 + -43.5t + 45 + t2 = 0 + 45 Reorder the terms: -45 + 45 + -43.5t + t2 = 0 + 45 Combine like terms: -45 + 45 = 0 0 + -43.5t + t2 = 0 + 45 -43.5t + t2 = 0 + 45 Combine like terms: 0 + 45 = 45 -43.5t + t2 = 45 The t term is -43.5t. Take half its coefficient (-21.75). Square it (473.0625) and add it to both sides. Add '473.0625' to each side of the equation. -43.5t + 473.0625 + t2 = 45 + 473.0625 Reorder the terms: 473.0625 + -43.5t + t2 = 45 + 473.0625 Combine like terms: 45 + 473.0625 = 518.0625 473.0625 + -43.5t + t2 = 518.0625 Factor a perfect square on the left side: (t + -21.75)(t + -21.75) = 518.0625 Calculate the square root of the right side: 22.760986358 Break this problem into two subproblems by setting (t + -21.75) equal to 22.760986358 and -22.760986358.Subproblem 1
t + -21.75 = 22.760986358 Simplifying t + -21.75 = 22.760986358 Reorder the terms: -21.75 + t = 22.760986358 Solving -21.75 + t = 22.760986358 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '21.75' to each side of the equation. -21.75 + 21.75 + t = 22.760986358 + 21.75 Combine like terms: -21.75 + 21.75 = 0.00 0.00 + t = 22.760986358 + 21.75 t = 22.760986358 + 21.75 Combine like terms: 22.760986358 + 21.75 = 44.510986358 t = 44.510986358 Simplifying t = 44.510986358Subproblem 2
t + -21.75 = -22.760986358 Simplifying t + -21.75 = -22.760986358 Reorder the terms: -21.75 + t = -22.760986358 Solving -21.75 + t = -22.760986358 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '21.75' to each side of the equation. -21.75 + 21.75 + t = -22.760986358 + 21.75 Combine like terms: -21.75 + 21.75 = 0.00 0.00 + t = -22.760986358 + 21.75 t = -22.760986358 + 21.75 Combine like terms: -22.760986358 + 21.75 = -1.010986358 t = -1.010986358 Simplifying t = -1.010986358Solution
The solution to the problem is based on the solutions from the subproblems. t = {44.510986358, -1.010986358}
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