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Simplifying 0 = 2t2 + -13t + -21 Reorder the terms: 0 = -21 + -13t + 2t2 Solving 0 = -21 + -13t + 2t2 Solving for variable 't'. Combine like terms: 0 + 21 = 21 21 + 13t + -2t2 = -21 + -13t + 2t2 + 21 + 13t + -2t2 Reorder the terms: 21 + 13t + -2t2 = -21 + 21 + -13t + 13t + 2t2 + -2t2 Combine like terms: -21 + 21 = 0 21 + 13t + -2t2 = 0 + -13t + 13t + 2t2 + -2t2 21 + 13t + -2t2 = -13t + 13t + 2t2 + -2t2 Combine like terms: -13t + 13t = 0 21 + 13t + -2t2 = 0 + 2t2 + -2t2 21 + 13t + -2t2 = 2t2 + -2t2 Combine like terms: 2t2 + -2t2 = 0 21 + 13t + -2t2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -10.5 + -6.5t + t2 = 0 Move the constant term to the right: Add '10.5' to each side of the equation. -10.5 + -6.5t + 10.5 + t2 = 0 + 10.5 Reorder the terms: -10.5 + 10.5 + -6.5t + t2 = 0 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + -6.5t + t2 = 0 + 10.5 -6.5t + t2 = 0 + 10.5 Combine like terms: 0 + 10.5 = 10.5 -6.5t + t2 = 10.5 The t term is -6.5t. Take half its coefficient (-3.25). Square it (10.5625) and add it to both sides. Add '10.5625' to each side of the equation. -6.5t + 10.5625 + t2 = 10.5 + 10.5625 Reorder the terms: 10.5625 + -6.5t + t2 = 10.5 + 10.5625 Combine like terms: 10.5 + 10.5625 = 21.0625 10.5625 + -6.5t + t2 = 21.0625 Factor a perfect square on the left side: (t + -3.25)(t + -3.25) = 21.0625 Calculate the square root of the right side: 4.589389938 Break this problem into two subproblems by setting (t + -3.25) equal to 4.589389938 and -4.589389938.Subproblem 1
t + -3.25 = 4.589389938 Simplifying t + -3.25 = 4.589389938 Reorder the terms: -3.25 + t = 4.589389938 Solving -3.25 + t = 4.589389938 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + t = 4.589389938 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + t = 4.589389938 + 3.25 t = 4.589389938 + 3.25 Combine like terms: 4.589389938 + 3.25 = 7.839389938 t = 7.839389938 Simplifying t = 7.839389938Subproblem 2
t + -3.25 = -4.589389938 Simplifying t + -3.25 = -4.589389938 Reorder the terms: -3.25 + t = -4.589389938 Solving -3.25 + t = -4.589389938 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + t = -4.589389938 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + t = -4.589389938 + 3.25 t = -4.589389938 + 3.25 Combine like terms: -4.589389938 + 3.25 = -1.339389938 t = -1.339389938 Simplifying t = -1.339389938Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.839389938, -1.339389938}
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