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Simplifying 0 = 8t2 + -25t + -78 Reorder the terms: 0 = -78 + -25t + 8t2 Solving 0 = -78 + -25t + 8t2 Solving for variable 't'. Combine like terms: 0 + 78 = 78 78 + 25t + -8t2 = -78 + -25t + 8t2 + 78 + 25t + -8t2 Reorder the terms: 78 + 25t + -8t2 = -78 + 78 + -25t + 25t + 8t2 + -8t2 Combine like terms: -78 + 78 = 0 78 + 25t + -8t2 = 0 + -25t + 25t + 8t2 + -8t2 78 + 25t + -8t2 = -25t + 25t + 8t2 + -8t2 Combine like terms: -25t + 25t = 0 78 + 25t + -8t2 = 0 + 8t2 + -8t2 78 + 25t + -8t2 = 8t2 + -8t2 Combine like terms: 8t2 + -8t2 = 0 78 + 25t + -8t2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -9.75 + -3.125t + t2 = 0 Move the constant term to the right: Add '9.75' to each side of the equation. -9.75 + -3.125t + 9.75 + t2 = 0 + 9.75 Reorder the terms: -9.75 + 9.75 + -3.125t + t2 = 0 + 9.75 Combine like terms: -9.75 + 9.75 = 0.00 0.00 + -3.125t + t2 = 0 + 9.75 -3.125t + t2 = 0 + 9.75 Combine like terms: 0 + 9.75 = 9.75 -3.125t + t2 = 9.75 The t term is -3.125t. Take half its coefficient (-1.5625). Square it (2.44140625) and add it to both sides. Add '2.44140625' to each side of the equation. -3.125t + 2.44140625 + t2 = 9.75 + 2.44140625 Reorder the terms: 2.44140625 + -3.125t + t2 = 9.75 + 2.44140625 Combine like terms: 9.75 + 2.44140625 = 12.19140625 2.44140625 + -3.125t + t2 = 12.19140625 Factor a perfect square on the left side: (t + -1.5625)(t + -1.5625) = 12.19140625 Calculate the square root of the right side: 3.491619431 Break this problem into two subproblems by setting (t + -1.5625) equal to 3.491619431 and -3.491619431.Subproblem 1
t + -1.5625 = 3.491619431 Simplifying t + -1.5625 = 3.491619431 Reorder the terms: -1.5625 + t = 3.491619431 Solving -1.5625 + t = 3.491619431 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5625' to each side of the equation. -1.5625 + 1.5625 + t = 3.491619431 + 1.5625 Combine like terms: -1.5625 + 1.5625 = 0.0000 0.0000 + t = 3.491619431 + 1.5625 t = 3.491619431 + 1.5625 Combine like terms: 3.491619431 + 1.5625 = 5.054119431 t = 5.054119431 Simplifying t = 5.054119431Subproblem 2
t + -1.5625 = -3.491619431 Simplifying t + -1.5625 = -3.491619431 Reorder the terms: -1.5625 + t = -3.491619431 Solving -1.5625 + t = -3.491619431 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.5625' to each side of the equation. -1.5625 + 1.5625 + t = -3.491619431 + 1.5625 Combine like terms: -1.5625 + 1.5625 = 0.0000 0.0000 + t = -3.491619431 + 1.5625 t = -3.491619431 + 1.5625 Combine like terms: -3.491619431 + 1.5625 = -1.929119431 t = -1.929119431 Simplifying t = -1.929119431Solution
The solution to the problem is based on the solutions from the subproblems. t = {5.054119431, -1.929119431}
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