33.5=t+5t^2

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Solution for 33.5=t+5t^2 equation: 


Simplifying
33.5 = t + 5t2

Solving
33.5 = t + 5t2

Solving for variable 't'.

Reorder the terms:
33.5 + -1t + -5t2 = t + -1t + 5t2 + -5t2

Combine like terms: t + -1t = 0
33.5 + -1t + -5t2 = 0 + 5t2 + -5t2
33.5 + -1t + -5t2 = 5t2 + -5t2

Combine like terms: 5t2 + -5t2 = 0
33.5 + -1t + -5t2 = 0

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

Divide each side by '-5'.
-6.7 + 0.2t + t2 = 0

Move the constant term to the right:

Add '6.7' to each side of the equation.
-6.7 + 0.2t + 6.7 + t2 = 0 + 6.7

Reorder the terms:
-6.7 + 6.7 + 0.2t + t2 = 0 + 6.7

Combine like terms: -6.7 + 6.7 = 0.0
0.0 + 0.2t + t2 = 0 + 6.7
0.2t + t2 = 0 + 6.7

Combine like terms: 0 + 6.7 = 6.7
0.2t + t2 = 6.7

The t term is 0.2t.  Take half its coefficient (0.1).
Square it (0.01) and add it to both sides.

Add '0.01' to each side of the equation.
0.2t + 0.01 + t2 = 6.7 + 0.01

Reorder the terms:
0.01 + 0.2t + t2 = 6.7 + 0.01

Combine like terms: 6.7 + 0.01 = 6.71
0.01 + 0.2t + t2 = 6.71

Factor a perfect square on the left side:
(t + 0.1)(t + 0.1) = 6.71

Calculate the square root of the right side: 2.590366769

Break this problem into two subproblems by setting 
(t + 0.1) equal to 2.590366769 and -2.590366769.


Subproblem 1
t + 0.1 = 2.590366769

Simplifying
t + 0.1 = 2.590366769

Reorder the terms:
0.1 + t = 2.590366769

Solving
0.1 + t = 2.590366769

Solving for variable 't'.

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

Add '-0.1' to each side of the equation.
0.1 + -0.1 + t = 2.590366769 + -0.1

Combine like terms: 0.1 + -0.1 = 0.0
0.0 + t = 2.590366769 + -0.1
t = 2.590366769 + -0.1

Combine like terms: 2.590366769 + -0.1 = 2.490366769
t = 2.490366769

Simplifying
t = 2.490366769


Subproblem 2
t + 0.1 = -2.590366769

Simplifying
t + 0.1 = -2.590366769

Reorder the terms:
0.1 + t = -2.590366769

Solving
0.1 + t = -2.590366769

Solving for variable 't'.

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

Add '-0.1' to each side of the equation.
0.1 + -0.1 + t = -2.590366769 + -0.1

Combine like terms: 0.1 + -0.1 = 0.0
0.0 + t = -2.590366769 + -0.1
t = -2.590366769 + -0.1

Combine like terms: -2.590366769 + -0.1 = -2.690366769
t = -2.690366769

Simplifying
t = -2.690366769


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {2.490366769, -2.690366769}

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