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Simplifying 16t2 + -58t + 31 = 0 Reorder the terms: 31 + -58t + 16t2 = 0 Solving 31 + -58t + 16t2 = 0 Solving for variable 't'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 1.9375 + -3.625t + t2 = 0 Move the constant term to the right: Add '-1.9375' to each side of the equation. 1.9375 + -3.625t + -1.9375 + t2 = 0 + -1.9375 Reorder the terms: 1.9375 + -1.9375 + -3.625t + t2 = 0 + -1.9375 Combine like terms: 1.9375 + -1.9375 = 0.0000 0.0000 + -3.625t + t2 = 0 + -1.9375 -3.625t + t2 = 0 + -1.9375 Combine like terms: 0 + -1.9375 = -1.9375 -3.625t + t2 = -1.9375 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 = -1.9375 + 3.28515625 Reorder the terms: 3.28515625 + -3.625t + t2 = -1.9375 + 3.28515625 Combine like terms: -1.9375 + 3.28515625 = 1.34765625 3.28515625 + -3.625t + t2 = 1.34765625 Factor a perfect square on the left side: (t + -1.8125)(t + -1.8125) = 1.34765625 Calculate the square root of the right side: 1.160885976 Break this problem into two subproblems by setting (t + -1.8125) equal to 1.160885976 and -1.160885976.Subproblem 1
t + -1.8125 = 1.160885976 Simplifying t + -1.8125 = 1.160885976 Reorder the terms: -1.8125 + t = 1.160885976 Solving -1.8125 + t = 1.160885976 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 = 1.160885976 + 1.8125 Combine like terms: -1.8125 + 1.8125 = 0.0000 0.0000 + t = 1.160885976 + 1.8125 t = 1.160885976 + 1.8125 Combine like terms: 1.160885976 + 1.8125 = 2.973385976 t = 2.973385976 Simplifying t = 2.973385976Subproblem 2
t + -1.8125 = -1.160885976 Simplifying t + -1.8125 = -1.160885976 Reorder the terms: -1.8125 + t = -1.160885976 Solving -1.8125 + t = -1.160885976 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 = -1.160885976 + 1.8125 Combine like terms: -1.8125 + 1.8125 = 0.0000 0.0000 + t = -1.160885976 + 1.8125 t = -1.160885976 + 1.8125 Combine like terms: -1.160885976 + 1.8125 = 0.651614024 t = 0.651614024 Simplifying t = 0.651614024Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.973385976, 0.651614024}
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