30=67t-5t^2

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Solution for 30=67t-5t^2 equation:


Simplifying
30 = 67t + -5t2

Solving
30 = 67t + -5t2

Solving for variable 't'.

Reorder the terms:
30 + -67t + 5t2 = 67t + -67t + -5t2 + 5t2

Combine like terms: 67t + -67t = 0
30 + -67t + 5t2 = 0 + -5t2 + 5t2
30 + -67t + 5t2 = -5t2 + 5t2

Combine like terms: -5t2 + 5t2 = 0
30 + -67t + 5t2 = 0

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

Divide each side by '5'.
6 + -13.4t + t2 = 0

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + -13.4t + -6 + t2 = 0 + -6

Reorder the terms:
6 + -6 + -13.4t + t2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -13.4t + t2 = 0 + -6
-13.4t + t2 = 0 + -6

Combine like terms: 0 + -6 = -6
-13.4t + t2 = -6

The t term is -13.4t.  Take half its coefficient (-6.7).
Square it (44.89) and add it to both sides.

Add '44.89' to each side of the equation.
-13.4t + 44.89 + t2 = -6 + 44.89

Reorder the terms:
44.89 + -13.4t + t2 = -6 + 44.89

Combine like terms: -6 + 44.89 = 38.89
44.89 + -13.4t + t2 = 38.89

Factor a perfect square on the left side:
(t + -6.7)(t + -6.7) = 38.89

Calculate the square root of the right side: 6.236184731

Break this problem into two subproblems by setting 
(t + -6.7) equal to 6.236184731 and -6.236184731.

Subproblem 1

t + -6.7 = 6.236184731 Simplifying t + -6.7 = 6.236184731 Reorder the terms: -6.7 + t = 6.236184731 Solving -6.7 + t = 6.236184731 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '6.7' to each side of the equation. -6.7 + 6.7 + t = 6.236184731 + 6.7 Combine like terms: -6.7 + 6.7 = 0.0 0.0 + t = 6.236184731 + 6.7 t = 6.236184731 + 6.7 Combine like terms: 6.236184731 + 6.7 = 12.936184731 t = 12.936184731 Simplifying t = 12.936184731

Subproblem 2

t + -6.7 = -6.236184731 Simplifying t + -6.7 = -6.236184731 Reorder the terms: -6.7 + t = -6.236184731 Solving -6.7 + t = -6.236184731 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '6.7' to each side of the equation. -6.7 + 6.7 + t = -6.236184731 + 6.7 Combine like terms: -6.7 + 6.7 = 0.0 0.0 + t = -6.236184731 + 6.7 t = -6.236184731 + 6.7 Combine like terms: -6.236184731 + 6.7 = 0.463815269 t = 0.463815269 Simplifying t = 0.463815269

Solution

The solution to the problem is based on the solutions from the subproblems. t = {12.936184731, 0.463815269}

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