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Simplifying 16t2 + 122t + 99 = 0 Reorder the terms: 99 + 122t + 16t2 = 0 Solving 99 + 122t + 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'. 6.1875 + 7.625t + t2 = 0 Move the constant term to the right: Add '-6.1875' to each side of the equation. 6.1875 + 7.625t + -6.1875 + t2 = 0 + -6.1875 Reorder the terms: 6.1875 + -6.1875 + 7.625t + t2 = 0 + -6.1875 Combine like terms: 6.1875 + -6.1875 = 0.0000 0.0000 + 7.625t + t2 = 0 + -6.1875 7.625t + t2 = 0 + -6.1875 Combine like terms: 0 + -6.1875 = -6.1875 7.625t + t2 = -6.1875 The t term is 7.625t. Take half its coefficient (3.8125). Square it (14.53515625) and add it to both sides. Add '14.53515625' to each side of the equation. 7.625t + 14.53515625 + t2 = -6.1875 + 14.53515625 Reorder the terms: 14.53515625 + 7.625t + t2 = -6.1875 + 14.53515625 Combine like terms: -6.1875 + 14.53515625 = 8.34765625 14.53515625 + 7.625t + t2 = 8.34765625 Factor a perfect square on the left side: (t + 3.8125)(t + 3.8125) = 8.34765625 Calculate the square root of the right side: 2.889231083 Break this problem into two subproblems by setting (t + 3.8125) equal to 2.889231083 and -2.889231083.Subproblem 1
t + 3.8125 = 2.889231083 Simplifying t + 3.8125 = 2.889231083 Reorder the terms: 3.8125 + t = 2.889231083 Solving 3.8125 + t = 2.889231083 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.8125' to each side of the equation. 3.8125 + -3.8125 + t = 2.889231083 + -3.8125 Combine like terms: 3.8125 + -3.8125 = 0.0000 0.0000 + t = 2.889231083 + -3.8125 t = 2.889231083 + -3.8125 Combine like terms: 2.889231083 + -3.8125 = -0.923268917 t = -0.923268917 Simplifying t = -0.923268917Subproblem 2
t + 3.8125 = -2.889231083 Simplifying t + 3.8125 = -2.889231083 Reorder the terms: 3.8125 + t = -2.889231083 Solving 3.8125 + t = -2.889231083 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-3.8125' to each side of the equation. 3.8125 + -3.8125 + t = -2.889231083 + -3.8125 Combine like terms: 3.8125 + -3.8125 = 0.0000 0.0000 + t = -2.889231083 + -3.8125 t = -2.889231083 + -3.8125 Combine like terms: -2.889231083 + -3.8125 = -6.701731083 t = -6.701731083 Simplifying t = -6.701731083Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.923268917, -6.701731083}
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