8t^2-67t+31=0

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Solution for 8t^2-67t+31=0 equation:


Simplifying
8t2 + -67t + 31 = 0

Reorder the terms:
31 + -67t + 8t2 = 0

Solving
31 + -67t + 8t2 = 0

Solving for variable 't'.

Begin completing the square.  Divide all terms by
8 the coefficient of the squared term: 

Divide each side by '8'.
3.875 + -8.375t + t2 = 0

Move the constant term to the right:

Add '-3.875' to each side of the equation.
3.875 + -8.375t + -3.875 + t2 = 0 + -3.875

Reorder the terms:
3.875 + -3.875 + -8.375t + t2 = 0 + -3.875

Combine like terms: 3.875 + -3.875 = 0.000
0.000 + -8.375t + t2 = 0 + -3.875
-8.375t + t2 = 0 + -3.875

Combine like terms: 0 + -3.875 = -3.875
-8.375t + t2 = -3.875

The t term is -8.375t.  Take half its coefficient (-4.1875).
Square it (17.53515625) and add it to both sides.

Add '17.53515625' to each side of the equation.
-8.375t + 17.53515625 + t2 = -3.875 + 17.53515625

Reorder the terms:
17.53515625 + -8.375t + t2 = -3.875 + 17.53515625

Combine like terms: -3.875 + 17.53515625 = 13.66015625
17.53515625 + -8.375t + t2 = 13.66015625

Factor a perfect square on the left side:
(t + -4.1875)(t + -4.1875) = 13.66015625

Calculate the square root of the right side: 3.69596486

Break this problem into two subproblems by setting 
(t + -4.1875) equal to 3.69596486 and -3.69596486.

Subproblem 1

t + -4.1875 = 3.69596486 Simplifying t + -4.1875 = 3.69596486 Reorder the terms: -4.1875 + t = 3.69596486 Solving -4.1875 + t = 3.69596486 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.1875' to each side of the equation. -4.1875 + 4.1875 + t = 3.69596486 + 4.1875 Combine like terms: -4.1875 + 4.1875 = 0.0000 0.0000 + t = 3.69596486 + 4.1875 t = 3.69596486 + 4.1875 Combine like terms: 3.69596486 + 4.1875 = 7.88346486 t = 7.88346486 Simplifying t = 7.88346486

Subproblem 2

t + -4.1875 = -3.69596486 Simplifying t + -4.1875 = -3.69596486 Reorder the terms: -4.1875 + t = -3.69596486 Solving -4.1875 + t = -3.69596486 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '4.1875' to each side of the equation. -4.1875 + 4.1875 + t = -3.69596486 + 4.1875 Combine like terms: -4.1875 + 4.1875 = 0.0000 0.0000 + t = -3.69596486 + 4.1875 t = -3.69596486 + 4.1875 Combine like terms: -3.69596486 + 4.1875 = 0.49153514 t = 0.49153514 Simplifying t = 0.49153514

Solution

The solution to the problem is based on the solutions from the subproblems. t = {7.88346486, 0.49153514}

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