6t(2t-3)-5(2t^2+9t-3)=

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Solution for 6t(2t-3)-5(2t^2+9t-3)= equation:


Simplifying
6t(2t + -3) + -5(2t2 + 9t + -3) = 0

Reorder the terms:
6t(-3 + 2t) + -5(2t2 + 9t + -3) = 0
(-3 * 6t + 2t * 6t) + -5(2t2 + 9t + -3) = 0
(-18t + 12t2) + -5(2t2 + 9t + -3) = 0

Reorder the terms:
-18t + 12t2 + -5(-3 + 9t + 2t2) = 0
-18t + 12t2 + (-3 * -5 + 9t * -5 + 2t2 * -5) = 0
-18t + 12t2 + (15 + -45t + -10t2) = 0

Reorder the terms:
15 + -18t + -45t + 12t2 + -10t2 = 0

Combine like terms: -18t + -45t = -63t
15 + -63t + 12t2 + -10t2 = 0

Combine like terms: 12t2 + -10t2 = 2t2
15 + -63t + 2t2 = 0

Solving
15 + -63t + 2t2 = 0

Solving for variable 't'.

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

Divide each side by '2'.
7.5 + -31.5t + t2 = 0

Move the constant term to the right:

Add '-7.5' to each side of the equation.
7.5 + -31.5t + -7.5 + t2 = 0 + -7.5

Reorder the terms:
7.5 + -7.5 + -31.5t + t2 = 0 + -7.5

Combine like terms: 7.5 + -7.5 = 0.0
0.0 + -31.5t + t2 = 0 + -7.5
-31.5t + t2 = 0 + -7.5

Combine like terms: 0 + -7.5 = -7.5
-31.5t + t2 = -7.5

The t term is -31.5t.  Take half its coefficient (-15.75).
Square it (248.0625) and add it to both sides.

Add '248.0625' to each side of the equation.
-31.5t + 248.0625 + t2 = -7.5 + 248.0625

Reorder the terms:
248.0625 + -31.5t + t2 = -7.5 + 248.0625

Combine like terms: -7.5 + 248.0625 = 240.5625
248.0625 + -31.5t + t2 = 240.5625

Factor a perfect square on the left side:
(t + -15.75)(t + -15.75) = 240.5625

Calculate the square root of the right side: 15.510077369

Break this problem into two subproblems by setting 
(t + -15.75) equal to 15.510077369 and -15.510077369.


Subproblem 1
t + -15.75 = 15.510077369

Simplifying
t + -15.75 = 15.510077369

Reorder the terms:
-15.75 + t = 15.510077369

Solving
-15.75 + t = 15.510077369

Solving for variable 't'.

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

Add '15.75' to each side of the equation.
-15.75 + 15.75 + t = 15.510077369 + 15.75

Combine like terms: -15.75 + 15.75 = 0.00
0.00 + t = 15.510077369 + 15.75
t = 15.510077369 + 15.75

Combine like terms: 15.510077369 + 15.75 = 31.260077369
t = 31.260077369

Simplifying
t = 31.260077369


Subproblem 2
t + -15.75 = -15.510077369

Simplifying
t + -15.75 = -15.510077369

Reorder the terms:
-15.75 + t = -15.510077369

Solving
-15.75 + t = -15.510077369

Solving for variable 't'.

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

Add '15.75' to each side of the equation.
-15.75 + 15.75 + t = -15.510077369 + 15.75

Combine like terms: -15.75 + 15.75 = 0.00
0.00 + t = -15.510077369 + 15.75
t = -15.510077369 + 15.75

Combine like terms: -15.510077369 + 15.75 = 0.239922631
t = 0.239922631

Simplifying
t = 0.239922631


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {31.260077369, 0.239922631}

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