4t(t-9)=6(t+7)

Solution for 4t(t-9)=6(t+7) equation:

Simplifying
4t(t + -9) = 6(t + 7)

Reorder the terms:
4t(-9 + t) = 6(t + 7)
(-9 * 4t + t * 4t) = 6(t + 7)
(-36t + 4t2) = 6(t + 7)

Reorder the terms:
-36t + 4t2 = 6(7 + t)
-36t + 4t2 = (7 * 6 + t * 6)
-36t + 4t2 = (42 + 6t)

Solving
-36t + 4t2 = 42 + 6t

Solving for variable 't'.

Reorder the terms:
-42 + -36t + -6t + 4t2 = 42 + 6t + -42 + -6t

Combine like terms: -36t + -6t = -42t
-42 + -42t + 4t2 = 42 + 6t + -42 + -6t

Reorder the terms:
-42 + -42t + 4t2 = 42 + -42 + 6t + -6t

Combine like terms: 42 + -42 = 0
-42 + -42t + 4t2 = 0 + 6t + -6t
-42 + -42t + 4t2 = 6t + -6t

Combine like terms: 6t + -6t = 0
-42 + -42t + 4t2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-21 + -21t + 2t2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-21 + -21t + 2t2)' equal to zero and attempt to solve:

Simplifying
-21 + -21t + 2t2 = 0

Solving
-21 + -21t + 2t2 = 0

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

Divide each side by '2'.
-10.5 + -10.5t + t2 = 0

Move the constant term to the right:

Add '10.5' to each side of the equation.
-10.5 + -10.5t + 10.5 + t2 = 0 + 10.5

Reorder the terms:
-10.5 + 10.5 + -10.5t + t2 = 0 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + -10.5t + t2 = 0 + 10.5
-10.5t + t2 = 0 + 10.5

Combine like terms: 0 + 10.5 = 10.5
-10.5t + t2 = 10.5

The t term is -10.5t.  Take half its coefficient (-5.25).
Square it (27.5625) and add it to both sides.

Add '27.5625' to each side of the equation.
-10.5t + 27.5625 + t2 = 10.5 + 27.5625

Reorder the terms:
27.5625 + -10.5t + t2 = 10.5 + 27.5625

Combine like terms: 10.5 + 27.5625 = 38.0625
27.5625 + -10.5t + t2 = 38.0625

Factor a perfect square on the left side:
(t + -5.25)(t + -5.25) = 38.0625

Calculate the square root of the right side: 6.16948134

Break this problem into two subproblems by setting 
(t + -5.25) equal to 6.16948134 and -6.16948134.

Subproblem 1
t + -5.25 = 6.16948134

Simplifying
t + -5.25 = 6.16948134

Reorder the terms:
-5.25 + t = 6.16948134

Solving
-5.25 + t = 6.16948134

Solving for variable 't'.

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

Add '5.25' to each side of the equation.
-5.25 + 5.25 + t = 6.16948134 + 5.25

Combine like terms: -5.25 + 5.25 = 0.00
0.00 + t = 6.16948134 + 5.25
t = 6.16948134 + 5.25

Combine like terms: 6.16948134 + 5.25 = 11.41948134
t = 11.41948134

Simplifying
t = 11.41948134

Subproblem 2
t + -5.25 = -6.16948134

Simplifying
t + -5.25 = -6.16948134

Reorder the terms:
-5.25 + t = -6.16948134

Solving
-5.25 + t = -6.16948134

Solving for variable 't'.

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

Add '5.25' to each side of the equation.
-5.25 + 5.25 + t = -6.16948134 + 5.25

Combine like terms: -5.25 + 5.25 = 0.00
0.00 + t = -6.16948134 + 5.25
t = -6.16948134 + 5.25

Combine like terms: -6.16948134 + 5.25 = -0.91948134
t = -0.91948134

Simplifying
t = -0.91948134

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {11.41948134, -0.91948134}
Solution
t = {11.41948134, -0.91948134}