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Simplifying 0 = 84t2 + -251t + 25 Reorder the terms: 0 = 25 + -251t + 84t2 Solving 0 = 25 + -251t + 84t2 Solving for variable 't'. Combine like terms: 0 + -25 = -25 -25 + 251t + -84t2 = 25 + -251t + 84t2 + -25 + 251t + -84t2 Reorder the terms: -25 + 251t + -84t2 = 25 + -25 + -251t + 251t + 84t2 + -84t2 Combine like terms: 25 + -25 = 0 -25 + 251t + -84t2 = 0 + -251t + 251t + 84t2 + -84t2 -25 + 251t + -84t2 = -251t + 251t + 84t2 + -84t2 Combine like terms: -251t + 251t = 0 -25 + 251t + -84t2 = 0 + 84t2 + -84t2 -25 + 251t + -84t2 = 84t2 + -84t2 Combine like terms: 84t2 + -84t2 = 0 -25 + 251t + -84t2 = 0 Begin completing the square. Divide all terms by -84 the coefficient of the squared term: Divide each side by '-84'. 0.2976190476 + -2.988095238t + t2 = 0 Move the constant term to the right: Add '-0.2976190476' to each side of the equation. 0.2976190476 + -2.988095238t + -0.2976190476 + t2 = 0 + -0.2976190476 Reorder the terms: 0.2976190476 + -0.2976190476 + -2.988095238t + t2 = 0 + -0.2976190476 Combine like terms: 0.2976190476 + -0.2976190476 = 0.0000000000 0.0000000000 + -2.988095238t + t2 = 0 + -0.2976190476 -2.988095238t + t2 = 0 + -0.2976190476 Combine like terms: 0 + -0.2976190476 = -0.2976190476 -2.988095238t + t2 = -0.2976190476 The t term is -2.988095238t. Take half its coefficient (-1.494047619). Square it (2.232178288) and add it to both sides. Add '2.232178288' to each side of the equation. -2.988095238t + 2.232178288 + t2 = -0.2976190476 + 2.232178288 Reorder the terms: 2.232178288 + -2.988095238t + t2 = -0.2976190476 + 2.232178288 Combine like terms: -0.2976190476 + 2.232178288 = 1.9345592404 2.232178288 + -2.988095238t + t2 = 1.9345592404 Factor a perfect square on the left side: (t + -1.494047619)(t + -1.494047619) = 1.9345592404 Calculate the square root of the right side: 1.390884338 Break this problem into two subproblems by setting (t + -1.494047619) equal to 1.390884338 and -1.390884338.Subproblem 1
t + -1.494047619 = 1.390884338 Simplifying t + -1.494047619 = 1.390884338 Reorder the terms: -1.494047619 + t = 1.390884338 Solving -1.494047619 + t = 1.390884338 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.494047619' to each side of the equation. -1.494047619 + 1.494047619 + t = 1.390884338 + 1.494047619 Combine like terms: -1.494047619 + 1.494047619 = 0.000000000 0.000000000 + t = 1.390884338 + 1.494047619 t = 1.390884338 + 1.494047619 Combine like terms: 1.390884338 + 1.494047619 = 2.884931957 t = 2.884931957 Simplifying t = 2.884931957Subproblem 2
t + -1.494047619 = -1.390884338 Simplifying t + -1.494047619 = -1.390884338 Reorder the terms: -1.494047619 + t = -1.390884338 Solving -1.494047619 + t = -1.390884338 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.494047619' to each side of the equation. -1.494047619 + 1.494047619 + t = -1.390884338 + 1.494047619 Combine like terms: -1.494047619 + 1.494047619 = 0.000000000 0.000000000 + t = -1.390884338 + 1.494047619 t = -1.390884338 + 1.494047619 Combine like terms: -1.390884338 + 1.494047619 = 0.103163281 t = 0.103163281 Simplifying t = 0.103163281Solution
The solution to the problem is based on the solutions from the subproblems. t = {2.884931957, 0.103163281}
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