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Simplifying 16t2 + 12.25t + -1000 = 0 Reorder the terms: -1000 + 12.25t + 16t2 = 0 Solving -1000 + 12.25t + 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'. -62.5 + 0.765625t + t2 = 0 Move the constant term to the right: Add '62.5' to each side of the equation. -62.5 + 0.765625t + 62.5 + t2 = 0 + 62.5 Reorder the terms: -62.5 + 62.5 + 0.765625t + t2 = 0 + 62.5 Combine like terms: -62.5 + 62.5 = 0.0 0.0 + 0.765625t + t2 = 0 + 62.5 0.765625t + t2 = 0 + 62.5 Combine like terms: 0 + 62.5 = 62.5 0.765625t + t2 = 62.5 The t term is 0.765625t. Take half its coefficient (0.3828125). Square it (0.1465454102) and add it to both sides. Add '0.1465454102' to each side of the equation. 0.765625t + 0.1465454102 + t2 = 62.5 + 0.1465454102 Reorder the terms: 0.1465454102 + 0.765625t + t2 = 62.5 + 0.1465454102 Combine like terms: 62.5 + 0.1465454102 = 62.6465454102 0.1465454102 + 0.765625t + t2 = 62.6465454102 Factor a perfect square on the left side: (t + 0.3828125)(t + 0.3828125) = 62.6465454102 Calculate the square root of the right side: 7.914957069 Break this problem into two subproblems by setting (t + 0.3828125) equal to 7.914957069 and -7.914957069.Subproblem 1
t + 0.3828125 = 7.914957069 Simplifying t + 0.3828125 = 7.914957069 Reorder the terms: 0.3828125 + t = 7.914957069 Solving 0.3828125 + t = 7.914957069 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3828125' to each side of the equation. 0.3828125 + -0.3828125 + t = 7.914957069 + -0.3828125 Combine like terms: 0.3828125 + -0.3828125 = 0.0000000 0.0000000 + t = 7.914957069 + -0.3828125 t = 7.914957069 + -0.3828125 Combine like terms: 7.914957069 + -0.3828125 = 7.532144569 t = 7.532144569 Simplifying t = 7.532144569Subproblem 2
t + 0.3828125 = -7.914957069 Simplifying t + 0.3828125 = -7.914957069 Reorder the terms: 0.3828125 + t = -7.914957069 Solving 0.3828125 + t = -7.914957069 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.3828125' to each side of the equation. 0.3828125 + -0.3828125 + t = -7.914957069 + -0.3828125 Combine like terms: 0.3828125 + -0.3828125 = 0.0000000 0.0000000 + t = -7.914957069 + -0.3828125 t = -7.914957069 + -0.3828125 Combine like terms: -7.914957069 + -0.3828125 = -8.297769569 t = -8.297769569 Simplifying t = -8.297769569Solution
The solution to the problem is based on the solutions from the subproblems. t = {7.532144569, -8.297769569}
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