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Simplifying 16t2 + 9t + -105 = 0 Reorder the terms: -105 + 9t + 16t2 = 0 Solving -105 + 9t + 16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -6.5625 + 0.5625t + t2 = 0 Move the constant term to the right: Add '6.5625' to each side of the equation. -6.5625 + 0.5625t + 6.5625 + t2 = 0 + 6.5625 Reorder the terms: -6.5625 + 6.5625 + 0.5625t + t2 = 0 + 6.5625 Combine like terms: -6.5625 + 6.5625 = 0.0000 0.0000 + 0.5625t + t2 = 0 + 6.5625 0.5625t + t2 = 0 + 6.5625 Combine like terms: 0 + 6.5625 = 6.5625 0.5625t + t2 = 6.5625 The t term is 0.5625t. Take half its coefficient (0.28125). Square it (0.0791015625) and add it to both sides. Add '0.0791015625' to each side of the equation. 0.5625t + 0.0791015625 + t2 = 6.5625 + 0.0791015625 Reorder the terms: 0.0791015625 + 0.5625t + t2 = 6.5625 + 0.0791015625 Combine like terms: 6.5625 + 0.0791015625 = 6.6416015625 0.0791015625 + 0.5625t + t2 = 6.6416015625 Factor a perfect square on the left side: (t + 0.28125)(t + 0.28125) = 6.6416015625 Calculate the square root of the right side: 2.57713049 Break this problem into two subproblems by setting (t + 0.28125) equal to 2.57713049 and -2.57713049.Subproblem 1
t + 0.28125 = 2.57713049 Simplifying t + 0.28125 = 2.57713049 Reorder the terms: 0.28125 + t = 2.57713049 Solving 0.28125 + t = 2.57713049 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.28125' to each side of the equation. 0.28125 + -0.28125 + t = 2.57713049 + -0.28125 Combine like terms: 0.28125 + -0.28125 = 0.00000 0.00000 + t = 2.57713049 + -0.28125 t = 2.57713049 + -0.28125 Combine like terms: 2.57713049 + -0.28125 = 2.29588049 t = 2.29588049 Simplifying t = 2.29588049Subproblem 2
t + 0.28125 = -2.57713049 Simplifying t + 0.28125 = -2.57713049 Reorder the terms: 0.28125 + t = -2.57713049 Solving 0.28125 + t = -2.57713049 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.28125' to each side of the equation. 0.28125 + -0.28125 + t = -2.57713049 + -0.28125 Combine like terms: 0.28125 + -0.28125 = 0.00000 0.00000 + t = -2.57713049 + -0.28125 t = -2.57713049 + -0.28125 Combine like terms: -2.57713049 + -0.28125 = -2.85838049 t = -2.85838049 Simplifying t = -2.85838049Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.29588049, -2.85838049}
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