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Simplifying 10t = 38 + -16t2 Solving 10t = 38 + -16t2 Solving for variable 't'. Reorder the terms: -38 + 10t + 16t2 = 38 + -16t2 + -38 + 16t2 Reorder the terms: -38 + 10t + 16t2 = 38 + -38 + -16t2 + 16t2 Combine like terms: 38 + -38 = 0 -38 + 10t + 16t2 = 0 + -16t2 + 16t2 -38 + 10t + 16t2 = -16t2 + 16t2 Combine like terms: -16t2 + 16t2 = 0 -38 + 10t + 16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-19 + 5t + 8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-19 + 5t + 8t2)' equal to zero and attempt to solve: Simplifying -19 + 5t + 8t2 = 0 Solving -19 + 5t + 8t2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -2.375 + 0.625t + t2 = 0 Move the constant term to the right: Add '2.375' to each side of the equation. -2.375 + 0.625t + 2.375 + t2 = 0 + 2.375 Reorder the terms: -2.375 + 2.375 + 0.625t + t2 = 0 + 2.375 Combine like terms: -2.375 + 2.375 = 0.000 0.000 + 0.625t + t2 = 0 + 2.375 0.625t + t2 = 0 + 2.375 Combine like terms: 0 + 2.375 = 2.375 0.625t + t2 = 2.375 The t term is 0.625t. Take half its coefficient (0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. 0.625t + 0.09765625 + t2 = 2.375 + 0.09765625 Reorder the terms: 0.09765625 + 0.625t + t2 = 2.375 + 0.09765625 Combine like terms: 2.375 + 0.09765625 = 2.47265625 0.09765625 + 0.625t + t2 = 2.47265625 Factor a perfect square on the left side: (t + 0.3125)(t + 0.3125) = 2.47265625 Calculate the square root of the right side: 1.572468203 Break this problem into two subproblems by setting (t + 0.3125) equal to 1.572468203 and -1.572468203.Subproblem 1
t + 0.3125 = 1.572468203 Simplifying t + 0.3125 = 1.572468203 Reorder the terms: 0.3125 + t = 1.572468203 Solving 0.3125 + t = 1.572468203 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + t = 1.572468203 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + t = 1.572468203 + -0.3125 t = 1.572468203 + -0.3125 Combine like terms: 1.572468203 + -0.3125 = 1.259968203 t = 1.259968203 Simplifying t = 1.259968203Subproblem 2
t + 0.3125 = -1.572468203 Simplifying t + 0.3125 = -1.572468203 Reorder the terms: 0.3125 + t = -1.572468203 Solving 0.3125 + t = -1.572468203 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + t = -1.572468203 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + t = -1.572468203 + -0.3125 t = -1.572468203 + -0.3125 Combine like terms: -1.572468203 + -0.3125 = -1.884968203 t = -1.884968203 Simplifying t = -1.884968203Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.259968203, -1.884968203}Solution
t = {1.259968203, -1.884968203}
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