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Simplifying 15 = 2t2 + -3t Reorder the terms: 15 = -3t + 2t2 Solving 15 = -3t + 2t2 Solving for variable 't'. Reorder the terms: 15 + 3t + -2t2 = -3t + 3t + 2t2 + -2t2 Combine like terms: -3t + 3t = 0 15 + 3t + -2t2 = 0 + 2t2 + -2t2 15 + 3t + -2t2 = 2t2 + -2t2 Combine like terms: 2t2 + -2t2 = 0 15 + 3t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -7.5 + -1.5t + t2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + -1.5t + 7.5 + t2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + -1.5t + t2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + -1.5t + t2 = 0 + 7.5 -1.5t + t2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 -1.5t + t2 = 7.5 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 = 7.5 + 0.5625 Reorder the terms: 0.5625 + -1.5t + t2 = 7.5 + 0.5625 Combine like terms: 7.5 + 0.5625 = 8.0625 0.5625 + -1.5t + t2 = 8.0625 Factor a perfect square on the left side: (t + -0.75)(t + -0.75) = 8.0625 Calculate the square root of the right side: 2.839454173 Break this problem into two subproblems by setting (t + -0.75) equal to 2.839454173 and -2.839454173.Subproblem 1
t + -0.75 = 2.839454173 Simplifying t + -0.75 = 2.839454173 Reorder the terms: -0.75 + t = 2.839454173 Solving -0.75 + t = 2.839454173 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 = 2.839454173 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = 2.839454173 + 0.75 t = 2.839454173 + 0.75 Combine like terms: 2.839454173 + 0.75 = 3.589454173 t = 3.589454173 Simplifying t = 3.589454173Subproblem 2
t + -0.75 = -2.839454173 Simplifying t + -0.75 = -2.839454173 Reorder the terms: -0.75 + t = -2.839454173 Solving -0.75 + t = -2.839454173 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 = -2.839454173 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + t = -2.839454173 + 0.75 t = -2.839454173 + 0.75 Combine like terms: -2.839454173 + 0.75 = -2.089454173 t = -2.089454173 Simplifying t = -2.089454173Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.589454173, -2.089454173}
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