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Simplifying 0 = 16t2 + 90t + 7 Reorder the terms: 0 = 7 + 90t + 16t2 Solving 0 = 7 + 90t + 16t2 Solving for variable 't'. Combine like terms: 0 + -7 = -7 -7 + -90t + -16t2 = 7 + 90t + 16t2 + -7 + -90t + -16t2 Reorder the terms: -7 + -90t + -16t2 = 7 + -7 + 90t + -90t + 16t2 + -16t2 Combine like terms: 7 + -7 = 0 -7 + -90t + -16t2 = 0 + 90t + -90t + 16t2 + -16t2 -7 + -90t + -16t2 = 90t + -90t + 16t2 + -16t2 Combine like terms: 90t + -90t = 0 -7 + -90t + -16t2 = 0 + 16t2 + -16t2 -7 + -90t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 -7 + -90t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(7 + 90t + 16t2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(7 + 90t + 16t2)' equal to zero and attempt to solve: Simplifying 7 + 90t + 16t2 = 0 Solving 7 + 90t + 16t2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.4375 + 5.625t + t2 = 0 Move the constant term to the right: Add '-0.4375' to each side of the equation. 0.4375 + 5.625t + -0.4375 + t2 = 0 + -0.4375 Reorder the terms: 0.4375 + -0.4375 + 5.625t + t2 = 0 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + 5.625t + t2 = 0 + -0.4375 5.625t + t2 = 0 + -0.4375 Combine like terms: 0 + -0.4375 = -0.4375 5.625t + t2 = -0.4375 The t term is 5.625t. Take half its coefficient (2.8125). Square it (7.91015625) and add it to both sides. Add '7.91015625' to each side of the equation. 5.625t + 7.91015625 + t2 = -0.4375 + 7.91015625 Reorder the terms: 7.91015625 + 5.625t + t2 = -0.4375 + 7.91015625 Combine like terms: -0.4375 + 7.91015625 = 7.47265625 7.91015625 + 5.625t + t2 = 7.47265625 Factor a perfect square on the left side: (t + 2.8125)(t + 2.8125) = 7.47265625 Calculate the square root of the right side: 2.733615966 Break this problem into two subproblems by setting (t + 2.8125) equal to 2.733615966 and -2.733615966.Subproblem 1
t + 2.8125 = 2.733615966 Simplifying t + 2.8125 = 2.733615966 Reorder the terms: 2.8125 + t = 2.733615966 Solving 2.8125 + t = 2.733615966 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.8125' to each side of the equation. 2.8125 + -2.8125 + t = 2.733615966 + -2.8125 Combine like terms: 2.8125 + -2.8125 = 0.0000 0.0000 + t = 2.733615966 + -2.8125 t = 2.733615966 + -2.8125 Combine like terms: 2.733615966 + -2.8125 = -0.078884034 t = -0.078884034 Simplifying t = -0.078884034Subproblem 2
t + 2.8125 = -2.733615966 Simplifying t + 2.8125 = -2.733615966 Reorder the terms: 2.8125 + t = -2.733615966 Solving 2.8125 + t = -2.733615966 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-2.8125' to each side of the equation. 2.8125 + -2.8125 + t = -2.733615966 + -2.8125 Combine like terms: 2.8125 + -2.8125 = 0.0000 0.0000 + t = -2.733615966 + -2.8125 t = -2.733615966 + -2.8125 Combine like terms: -2.733615966 + -2.8125 = -5.546115966 t = -5.546115966 Simplifying t = -5.546115966Solution
The solution to the problem is based on the solutions from the subproblems. t = {-0.078884034, -5.546115966}Solution
t = {-0.078884034, -5.546115966}
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