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Simplifying 6(58) = 16t2 + 58t + 6 Multiply 6 * 58 348 = 16t2 + 58t + 6 Reorder the terms: 348 = 6 + 58t + 16t2 Solving 348 = 6 + 58t + 16t2 Solving for variable 't'. Combine like terms: 348 + -6 = 342 342 + -58t + -16t2 = 6 + 58t + 16t2 + -6 + -58t + -16t2 Reorder the terms: 342 + -58t + -16t2 = 6 + -6 + 58t + -58t + 16t2 + -16t2 Combine like terms: 6 + -6 = 0 342 + -58t + -16t2 = 0 + 58t + -58t + 16t2 + -16t2 342 + -58t + -16t2 = 58t + -58t + 16t2 + -16t2 Combine like terms: 58t + -58t = 0 342 + -58t + -16t2 = 0 + 16t2 + -16t2 342 + -58t + -16t2 = 16t2 + -16t2 Combine like terms: 16t2 + -16t2 = 0 342 + -58t + -16t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(171 + -29t + -8t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(171 + -29t + -8t2)' equal to zero and attempt to solve: Simplifying 171 + -29t + -8t2 = 0 Solving 171 + -29t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -21.375 + 3.625t + t2 = 0 Move the constant term to the right: Add '21.375' to each side of the equation. -21.375 + 3.625t + 21.375 + t2 = 0 + 21.375 Reorder the terms: -21.375 + 21.375 + 3.625t + t2 = 0 + 21.375 Combine like terms: -21.375 + 21.375 = 0.000 0.000 + 3.625t + t2 = 0 + 21.375 3.625t + t2 = 0 + 21.375 Combine like terms: 0 + 21.375 = 21.375 3.625t + t2 = 21.375 The t term is 3.625t. Take half its coefficient (1.8125). Square it (3.28515625) and add it to both sides. Add '3.28515625' to each side of the equation. 3.625t + 3.28515625 + t2 = 21.375 + 3.28515625 Reorder the terms: 3.28515625 + 3.625t + t2 = 21.375 + 3.28515625 Combine like terms: 21.375 + 3.28515625 = 24.66015625 3.28515625 + 3.625t + t2 = 24.66015625 Factor a perfect square on the left side: (t + 1.8125)(t + 1.8125) = 24.66015625 Calculate the square root of the right side: 4.965899339 Break this problem into two subproblems by setting (t + 1.8125) equal to 4.965899339 and -4.965899339.Subproblem 1
t + 1.8125 = 4.965899339 Simplifying t + 1.8125 = 4.965899339 Reorder the terms: 1.8125 + t = 4.965899339 Solving 1.8125 + t = 4.965899339 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.8125' to each side of the equation. 1.8125 + -1.8125 + t = 4.965899339 + -1.8125 Combine like terms: 1.8125 + -1.8125 = 0.0000 0.0000 + t = 4.965899339 + -1.8125 t = 4.965899339 + -1.8125 Combine like terms: 4.965899339 + -1.8125 = 3.153399339 t = 3.153399339 Simplifying t = 3.153399339Subproblem 2
t + 1.8125 = -4.965899339 Simplifying t + 1.8125 = -4.965899339 Reorder the terms: 1.8125 + t = -4.965899339 Solving 1.8125 + t = -4.965899339 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.8125' to each side of the equation. 1.8125 + -1.8125 + t = -4.965899339 + -1.8125 Combine like terms: 1.8125 + -1.8125 = 0.0000 0.0000 + t = -4.965899339 + -1.8125 t = -4.965899339 + -1.8125 Combine like terms: -4.965899339 + -1.8125 = -6.778399339 t = -6.778399339 Simplifying t = -6.778399339Solution
The solution to the problem is based on the solutions from the subproblems. t = {3.153399339, -6.778399339}Solution
t = {3.153399339, -6.778399339}
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