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Simplifying (-16t2) + 24t + 9 = 18t Reorder the terms: 9 + 24t + (-16t2) = 18t Solving 9 + 24t + (-16t2) = 18t Solving for variable 't'. Reorder the terms: 9 + 24t + -18t + (-16t2) = 18t + -18t Combine like terms: 24t + -18t = 6t 9 + 6t + (-16t2) = 18t + -18t Combine like terms: 18t + -18t = 0 9 + 6t + (-16t2) = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.5625 + -0.375t + t2 = 0 Move the constant term to the right: Add '0.5625' to each side of the equation. -0.5625 + -0.375t + 0.5625 + t2 = 0 + 0.5625 Reorder the terms: -0.5625 + 0.5625 + -0.375t + t2 = 0 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + -0.375t + t2 = 0 + 0.5625 -0.375t + t2 = 0 + 0.5625 Combine like terms: 0 + 0.5625 = 0.5625 -0.375t + t2 = 0.5625 The t term is -0.375t. Take half its coefficient (-0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. -0.375t + 0.03515625 + t2 = 0.5625 + 0.03515625 Reorder the terms: 0.03515625 + -0.375t + t2 = 0.5625 + 0.03515625 Combine like terms: 0.5625 + 0.03515625 = 0.59765625 0.03515625 + -0.375t + t2 = 0.59765625 Factor a perfect square on the left side: ((t) + -0.1875)((t) + -0.1875) = 0.59765625 Calculate the square root of the right side: 0.773082305 Break this problem into two subproblems by setting ((t) + -0.1875) equal to 0.773082305 and -0.773082305.Subproblem 1
(t) + -0.1875 = 0.773082305 Simplifying (t) + -0.1875 = 0.773082305 t + -0.1875 = 0.773082305 Reorder the terms: -0.1875 + t = 0.773082305 Solving -0.1875 + t = 0.773082305 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + t = 0.773082305 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + t = 0.773082305 + 0.1875 t = 0.773082305 + 0.1875 Combine like terms: 0.773082305 + 0.1875 = 0.960582305 t = 0.960582305 Simplifying t = 0.960582305Subproblem 2
(t) + -0.1875 = -0.773082305 Simplifying (t) + -0.1875 = -0.773082305 t + -0.1875 = -0.773082305 Reorder the terms: -0.1875 + t = -0.773082305 Solving -0.1875 + t = -0.773082305 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + t = -0.773082305 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + t = -0.773082305 + 0.1875 t = -0.773082305 + 0.1875 Combine like terms: -0.773082305 + 0.1875 = -0.585582305 t = -0.585582305 Simplifying t = -0.585582305Solution
The solution to the problem is based on the solutions from the subproblems. t = {0.960582305, -0.585582305}
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