t^2+12t=84

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Solution for t^2+12t=84 equation:


Simplifying
t2 + 12t = 84

Reorder the terms:
12t + t2 = 84

Solving
12t + t2 = 84

Solving for variable 't'.

Reorder the terms:
-84 + 12t + t2 = 84 + -84

Combine like terms: 84 + -84 = 0
-84 + 12t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '84' to each side of the equation.
-84 + 12t + 84 + t2 = 0 + 84

Reorder the terms:
-84 + 84 + 12t + t2 = 0 + 84

Combine like terms: -84 + 84 = 0
0 + 12t + t2 = 0 + 84
12t + t2 = 0 + 84

Combine like terms: 0 + 84 = 84
12t + t2 = 84

The t term is 12t.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12t + 36 + t2 = 84 + 36

Reorder the terms:
36 + 12t + t2 = 84 + 36

Combine like terms: 84 + 36 = 120
36 + 12t + t2 = 120

Factor a perfect square on the left side:
(t + 6)(t + 6) = 120

Calculate the square root of the right side: 10.95445115

Break this problem into two subproblems by setting 
(t + 6) equal to 10.95445115 and -10.95445115.

Subproblem 1

t + 6 = 10.95445115 Simplifying t + 6 = 10.95445115 Reorder the terms: 6 + t = 10.95445115 Solving 6 + t = 10.95445115 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + t = 10.95445115 + -6 Combine like terms: 6 + -6 = 0 0 + t = 10.95445115 + -6 t = 10.95445115 + -6 Combine like terms: 10.95445115 + -6 = 4.95445115 t = 4.95445115 Simplifying t = 4.95445115

Subproblem 2

t + 6 = -10.95445115 Simplifying t + 6 = -10.95445115 Reorder the terms: 6 + t = -10.95445115 Solving 6 + t = -10.95445115 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + t = -10.95445115 + -6 Combine like terms: 6 + -6 = 0 0 + t = -10.95445115 + -6 t = -10.95445115 + -6 Combine like terms: -10.95445115 + -6 = -16.95445115 t = -16.95445115 Simplifying t = -16.95445115

Solution

The solution to the problem is based on the solutions from the subproblems. t = {4.95445115, -16.95445115}

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