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Simplifying 16t2 + -70t + 45 = 0 Reorder the terms: 45 + -70t + 16t2 = 0 Solving 45 + -70t + 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'. 2.8125 + -4.375t + t2 = 0 Move the constant term to the right: Add '-2.8125' to each side of the equation. 2.8125 + -4.375t + -2.8125 + t2 = 0 + -2.8125 Reorder the terms: 2.8125 + -2.8125 + -4.375t + t2 = 0 + -2.8125 Combine like terms: 2.8125 + -2.8125 = 0.0000 0.0000 + -4.375t + t2 = 0 + -2.8125 -4.375t + t2 = 0 + -2.8125 Combine like terms: 0 + -2.8125 = -2.8125 -4.375t + t2 = -2.8125 The t term is -4.375t. Take half its coefficient (-2.1875). Square it (4.78515625) and add it to both sides. Add '4.78515625' to each side of the equation. -4.375t + 4.78515625 + t2 = -2.8125 + 4.78515625 Reorder the terms: 4.78515625 + -4.375t + t2 = -2.8125 + 4.78515625 Combine like terms: -2.8125 + 4.78515625 = 1.97265625 4.78515625 + -4.375t + t2 = 1.97265625 Factor a perfect square on the left side: (t + -2.1875)(t + -2.1875) = 1.97265625 Calculate the square root of the right side: 1.404512816 Break this problem into two subproblems by setting (t + -2.1875) equal to 1.404512816 and -1.404512816.Subproblem 1
t + -2.1875 = 1.404512816 Simplifying t + -2.1875 = 1.404512816 Reorder the terms: -2.1875 + t = 1.404512816 Solving -2.1875 + t = 1.404512816 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.1875' to each side of the equation. -2.1875 + 2.1875 + t = 1.404512816 + 2.1875 Combine like terms: -2.1875 + 2.1875 = 0.0000 0.0000 + t = 1.404512816 + 2.1875 t = 1.404512816 + 2.1875 Combine like terms: 1.404512816 + 2.1875 = 3.592012816 t = 3.592012816 Simplifying t = 3.592012816Subproblem 2
t + -2.1875 = -1.404512816 Simplifying t + -2.1875 = -1.404512816 Reorder the terms: -2.1875 + t = -1.404512816 Solving -2.1875 + t = -1.404512816 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.1875' to each side of the equation. -2.1875 + 2.1875 + t = -1.404512816 + 2.1875 Combine like terms: -2.1875 + 2.1875 = 0.0000 0.0000 + t = -1.404512816 + 2.1875 t = -1.404512816 + 2.1875 Combine like terms: -1.404512816 + 2.1875 = 0.782987184 t = 0.782987184 Simplifying t = 0.782987184Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.592012816, 0.782987184}
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