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Simplifying 70 = -16t2 + 70t + 5 Reorder the terms: 70 = 5 + 70t + -16t2 Solving 70 = 5 + 70t + -16t2 Solving for variable 't'. Combine like terms: 70 + -5 = 65 65 + -70t + 16t2 = 5 + 70t + -16t2 + -5 + -70t + 16t2 Reorder the terms: 65 + -70t + 16t2 = 5 + -5 + 70t + -70t + -16t2 + 16t2 Combine like terms: 5 + -5 = 0 65 + -70t + 16t2 = 0 + 70t + -70t + -16t2 + 16t2 65 + -70t + 16t2 = 70t + -70t + -16t2 + 16t2 Combine like terms: 70t + -70t = 0 65 + -70t + 16t2 = 0 + -16t2 + 16t2 65 + -70t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 65 + -70t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 4.0625 + -4.375t + t2 = 0 Move the constant term to the right: Add '-4.0625' to each side of the equation. 4.0625 + -4.375t + -4.0625 + t2 = 0 + -4.0625 Reorder the terms: 4.0625 + -4.0625 + -4.375t + t2 = 0 + -4.0625 Combine like terms: 4.0625 + -4.0625 = 0.0000 0.0000 + -4.375t + t2 = 0 + -4.0625 -4.375t + t2 = 0 + -4.0625 Combine like terms: 0 + -4.0625 = -4.0625 -4.375t + t2 = -4.0625 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 = -4.0625 + 4.78515625 Reorder the terms: 4.78515625 + -4.375t + t2 = -4.0625 + 4.78515625 Combine like terms: -4.0625 + 4.78515625 = 0.72265625 4.78515625 + -4.375t + t2 = 0.72265625 Factor a perfect square on the left side: (t + -2.1875)(t + -2.1875) = 0.72265625 Calculate the square root of the right side: 0.850091907 Break this problem into two subproblems by setting (t + -2.1875) equal to 0.850091907 and -0.850091907.Subproblem 1
t + -2.1875 = 0.850091907 Simplifying t + -2.1875 = 0.850091907 Reorder the terms: -2.1875 + t = 0.850091907 Solving -2.1875 + t = 0.850091907 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 = 0.850091907 + 2.1875 Combine like terms: -2.1875 + 2.1875 = 0.0000 0.0000 + t = 0.850091907 + 2.1875 t = 0.850091907 + 2.1875 Combine like terms: 0.850091907 + 2.1875 = 3.037591907 t = 3.037591907 Simplifying t = 3.037591907Subproblem 2
t + -2.1875 = -0.850091907 Simplifying t + -2.1875 = -0.850091907 Reorder the terms: -2.1875 + t = -0.850091907 Solving -2.1875 + t = -0.850091907 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 = -0.850091907 + 2.1875 Combine like terms: -2.1875 + 2.1875 = 0.0000 0.0000 + t = -0.850091907 + 2.1875 t = -0.850091907 + 2.1875 Combine like terms: -0.850091907 + 2.1875 = 1.337408093 t = 1.337408093 Simplifying t = 1.337408093Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.037591907, 1.337408093}
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