t(9t+11)=6.5

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Solution for t(9t+11)=6.5 equation:


Simplifying
t(9t + 11) = 6.5

Reorder the terms:
t(11 + 9t) = 6.5
(11 * t + 9t * t) = 6.5
(11t + 9t2) = 6.5

Solving
11t + 9t2 = 6.5

Solving for variable 't'.

Reorder the terms:
-6.5 + 11t + 9t2 = 6.5 + -6.5

Combine like terms: 6.5 + -6.5 = 0.0
-6.5 + 11t + 9t2 = 0.0

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

Divide each side by '9'.
-0.7222222222 + 1.222222222t + t2 = 0.0

Move the constant term to the right:

Add '0.7222222222' to each side of the equation.
-0.7222222222 + 1.222222222t + 0.7222222222 + t2 = 0.0 + 0.7222222222

Reorder the terms:
-0.7222222222 + 0.7222222222 + 1.222222222t + t2 = 0.0 + 0.7222222222

Combine like terms: -0.7222222222 + 0.7222222222 = 0.0000000000
0.0000000000 + 1.222222222t + t2 = 0.0 + 0.7222222222
1.222222222t + t2 = 0.0 + 0.7222222222

Combine like terms: 0.0 + 0.7222222222 = 0.7222222222
1.222222222t + t2 = 0.7222222222

The t term is 1.222222222t.  Take half its coefficient (0.611111111).
Square it (0.3734567900) and add it to both sides.

Add '0.3734567900' to each side of the equation.
1.222222222t + 0.3734567900 + t2 = 0.7222222222 + 0.3734567900

Reorder the terms:
0.3734567900 + 1.222222222t + t2 = 0.7222222222 + 0.3734567900

Combine like terms: 0.7222222222 + 0.3734567900 = 1.0956790122
0.3734567900 + 1.222222222t + t2 = 1.0956790122

Factor a perfect square on the left side:
(t + 0.611111111)(t + 0.611111111) = 1.0956790122

Calculate the square root of the right side: 1.046746871

Break this problem into two subproblems by setting 
(t + 0.611111111) equal to 1.046746871 and -1.046746871.

Subproblem 1

t + 0.611111111 = 1.046746871 Simplifying t + 0.611111111 = 1.046746871 Reorder the terms: 0.611111111 + t = 1.046746871 Solving 0.611111111 + t = 1.046746871 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.611111111' to each side of the equation. 0.611111111 + -0.611111111 + t = 1.046746871 + -0.611111111 Combine like terms: 0.611111111 + -0.611111111 = 0.000000000 0.000000000 + t = 1.046746871 + -0.611111111 t = 1.046746871 + -0.611111111 Combine like terms: 1.046746871 + -0.611111111 = 0.43563576 t = 0.43563576 Simplifying t = 0.43563576

Subproblem 2

t + 0.611111111 = -1.046746871 Simplifying t + 0.611111111 = -1.046746871 Reorder the terms: 0.611111111 + t = -1.046746871 Solving 0.611111111 + t = -1.046746871 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-0.611111111' to each side of the equation. 0.611111111 + -0.611111111 + t = -1.046746871 + -0.611111111 Combine like terms: 0.611111111 + -0.611111111 = 0.000000000 0.000000000 + t = -1.046746871 + -0.611111111 t = -1.046746871 + -0.611111111 Combine like terms: -1.046746871 + -0.611111111 = -1.657857982 t = -1.657857982 Simplifying t = -1.657857982

Solution

The solution to the problem is based on the solutions from the subproblems. t = {0.43563576, -1.657857982}

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