0=t^2+9t+10

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Solution for 0=t^2+9t+10 equation: 


Simplifying
0 = t2 + 9t + 10

Reorder the terms:
0 = 10 + 9t + t2

Solving
0 = 10 + 9t + t2

Solving for variable 't'.

Combine like terms: 0 + -10 = -10
-10 + -9t + -1t2 = 10 + 9t + t2 + -10 + -9t + -1t2

Reorder the terms:
-10 + -9t + -1t2 = 10 + -10 + 9t + -9t + t2 + -1t2

Combine like terms: 10 + -10 = 0
-10 + -9t + -1t2 = 0 + 9t + -9t + t2 + -1t2
-10 + -9t + -1t2 = 9t + -9t + t2 + -1t2

Combine like terms: 9t + -9t = 0
-10 + -9t + -1t2 = 0 + t2 + -1t2
-10 + -9t + -1t2 = t2 + -1t2

Combine like terms: t2 + -1t2 = 0
-10 + -9t + -1t2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(10 + 9t + t2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(10 + 9t + t2)' equal to zero and attempt to solve:

Simplifying
10 + 9t + t2 = 0

Solving
10 + 9t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-10' to each side of the equation.
10 + 9t + -10 + t2 = 0 + -10

Reorder the terms:
10 + -10 + 9t + t2 = 0 + -10

Combine like terms: 10 + -10 = 0
0 + 9t + t2 = 0 + -10
9t + t2 = 0 + -10

Combine like terms: 0 + -10 = -10
9t + t2 = -10

The t term is 9t.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9t + 20.25 + t2 = -10 + 20.25

Reorder the terms:
20.25 + 9t + t2 = -10 + 20.25

Combine like terms: -10 + 20.25 = 10.25
20.25 + 9t + t2 = 10.25

Factor a perfect square on the left side:
(t + 4.5)(t + 4.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(t + 4.5) equal to 3.201562119 and -3.201562119.


Subproblem 1
t + 4.5 = 3.201562119

Simplifying
t + 4.5 = 3.201562119

Reorder the terms:
4.5 + t = 3.201562119

Solving
4.5 + t = 3.201562119

Solving for variable 't'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + t = 3.201562119 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + t = 3.201562119 + -4.5
t = 3.201562119 + -4.5

Combine like terms: 3.201562119 + -4.5 = -1.298437881
t = -1.298437881

Simplifying
t = -1.298437881


Subproblem 2
t + 4.5 = -3.201562119

Simplifying
t + 4.5 = -3.201562119

Reorder the terms:
4.5 + t = -3.201562119

Solving
4.5 + t = -3.201562119

Solving for variable 't'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + t = -3.201562119 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + t = -3.201562119 + -4.5
t = -3.201562119 + -4.5

Combine like terms: -3.201562119 + -4.5 = -7.701562119
t = -7.701562119

Simplifying
t = -7.701562119


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {-1.298437881, -7.701562119}
Solution
t = {-1.298437881, -7.701562119}

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