t^2+16t+78=59

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Solution for t^2+16t+78=59 equation: 


Simplifying
t2 + 16t + 78 = 59

Reorder the terms:
78 + 16t + t2 = 59

Solving
78 + 16t + t2 = 59

Solving for variable 't'.

Reorder the terms:
78 + -59 + 16t + t2 = 59 + -59

Combine like terms: 78 + -59 = 19
19 + 16t + t2 = 59 + -59

Combine like terms: 59 + -59 = 0
19 + 16t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-19' to each side of the equation.
19 + 16t + -19 + t2 = 0 + -19

Reorder the terms:
19 + -19 + 16t + t2 = 0 + -19

Combine like terms: 19 + -19 = 0
0 + 16t + t2 = 0 + -19
16t + t2 = 0 + -19

Combine like terms: 0 + -19 = -19
16t + t2 = -19

The t term is 16t.  Take half its coefficient (8).
Square it (64) and add it to both sides.

Add '64' to each side of the equation.
16t + 64 + t2 = -19 + 64

Reorder the terms:
64 + 16t + t2 = -19 + 64

Combine like terms: -19 + 64 = 45
64 + 16t + t2 = 45

Factor a perfect square on the left side:
(t + 8)(t + 8) = 45

Calculate the square root of the right side: 6.708203933

Break this problem into two subproblems by setting 
(t + 8) equal to 6.708203933 and -6.708203933.


Subproblem 1
t + 8 = 6.708203933

Simplifying
t + 8 = 6.708203933

Reorder the terms:
8 + t = 6.708203933

Solving
8 + t = 6.708203933

Solving for variable 't'.

Move all terms containing t to the left, all other terms to the right.

Add '-8' to each side of the equation.
8 + -8 + t = 6.708203933 + -8

Combine like terms: 8 + -8 = 0
0 + t = 6.708203933 + -8
t = 6.708203933 + -8

Combine like terms: 6.708203933 + -8 = -1.291796067
t = -1.291796067

Simplifying
t = -1.291796067


Subproblem 2
t + 8 = -6.708203933

Simplifying
t + 8 = -6.708203933

Reorder the terms:
8 + t = -6.708203933

Solving
8 + t = -6.708203933

Solving for variable 't'.

Move all terms containing t to the left, all other terms to the right.

Add '-8' to each side of the equation.
8 + -8 + t = -6.708203933 + -8

Combine like terms: 8 + -8 = 0
0 + t = -6.708203933 + -8
t = -6.708203933 + -8

Combine like terms: -6.708203933 + -8 = -14.708203933
t = -14.708203933

Simplifying
t = -14.708203933


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {-1.291796067, -14.708203933}

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