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Simplifying -5 = 16t2 + 70t + 4 Reorder the terms: -5 = 4 + 70t + 16t2 Solving -5 = 4 + 70t + 16t2 Solving for variable 't'. Combine like terms: -5 + -4 = -9 -9 + -70t + -16t2 = 4 + 70t + 16t2 + -4 + -70t + -16t2 Reorder the terms: -9 + -70t + -16t2 = 4 + -4 + 70t + -70t + 16t2 + -16t2 Combine like terms: 4 + -4 = 0 -9 + -70t + -16t2 = 0 + 70t + -70t + 16t2 + -16t2 -9 + -70t + -16t2 = 70t + -70t + 16t2 + -16t2 Combine like terms: 70t + -70t = 0 -9 + -70t + -16t2 = 0 + 16t2 + -16t2 -9 + -70t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -9 + -70t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(9 + 70t + 16t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(9 + 70t + 16t2)' equal to zero and attempt to solve: Simplifying 9 + 70t + 16t2 = 0 Solving 9 + 70t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.5625 + 4.375t + t2 = 0 Move the constant term to the right: Add '-0.5625' to each side of the equation. 0.5625 + 4.375t + -0.5625 + t2 = 0 + -0.5625 Reorder the terms: 0.5625 + -0.5625 + 4.375t + t2 = 0 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + 4.375t + t2 = 0 + -0.5625 4.375t + t2 = 0 + -0.5625 Combine like terms: 0 + -0.5625 = -0.5625 4.375t + t2 = -0.5625 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 = -0.5625 + 4.78515625 Reorder the terms: 4.78515625 + 4.375t + t2 = -0.5625 + 4.78515625 Combine like terms: -0.5625 + 4.78515625 = 4.22265625 4.78515625 + 4.375t + t2 = 4.22265625 Factor a perfect square on the left side: (t + 2.1875)(t + 2.1875) = 4.22265625 Calculate the square root of the right side: 2.054910278 Break this problem into two subproblems by setting (t + 2.1875) equal to 2.054910278 and -2.054910278.Subproblem 1
t + 2.1875 = 2.054910278 Simplifying t + 2.1875 = 2.054910278 Reorder the terms: 2.1875 + t = 2.054910278 Solving 2.1875 + t = 2.054910278 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 = 2.054910278 + -2.1875 Combine like terms: 2.1875 + -2.1875 = 0.0000 0.0000 + t = 2.054910278 + -2.1875 t = 2.054910278 + -2.1875 Combine like terms: 2.054910278 + -2.1875 = -0.132589722 t = -0.132589722 Simplifying t = -0.132589722Subproblem 2
t + 2.1875 = -2.054910278 Simplifying t + 2.1875 = -2.054910278 Reorder the terms: 2.1875 + t = -2.054910278 Solving 2.1875 + t = -2.054910278 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 = -2.054910278 + -2.1875 Combine like terms: 2.1875 + -2.1875 = 0.0000 0.0000 + t = -2.054910278 + -2.1875 t = -2.054910278 + -2.1875 Combine like terms: -2.054910278 + -2.1875 = -4.242410278 t = -4.242410278 Simplifying t = -4.242410278Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.132589722, -4.242410278}Solution
t = {-0.132589722, -4.242410278}
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