t^3+3t=7t^2

Solution for t^3+3t=7t^2 equation:

Simplifying
t3 + 3t = 7t2

Reorder the terms:
3t + t3 = 7t2

Solving
3t + t3 = 7t2

Solving for variable 't'.

Reorder the terms:
3t + -7t2 + t3 = 7t2 + -7t2

Combine like terms: 7t2 + -7t2 = 0
3t + -7t2 + t3 = 0

Factor out the Greatest Common Factor (GCF), 't'.
t(3 + -7t + t2) = 0

Subproblem 1
Set the factor 't' equal to zero and attempt to solve:

Simplifying
t = 0

Solving
t = 0

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

Simplifying
t = 0
Subproblem 2
Set the factor '(3 + -7t + t2)' equal to zero and attempt to solve:

Simplifying
3 + -7t + t2 = 0

Solving
3 + -7t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -7t + -3 + t2 = 0 + -3

Reorder the terms:
3 + -3 + -7t + t2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -7t + t2 = 0 + -3
-7t + t2 = 0 + -3

Combine like terms: 0 + -3 = -3
-7t + t2 = -3

The t term is -7t.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7t + 12.25 + t2 = -3 + 12.25

Reorder the terms:
12.25 + -7t + t2 = -3 + 12.25

Combine like terms: -3 + 12.25 = 9.25
12.25 + -7t + t2 = 9.25

Factor a perfect square on the left side:
(t + -3.5)(t + -3.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(t + -3.5) equal to 3.041381265 and -3.041381265.

Subproblem 1
t + -3.5 = 3.041381265

Simplifying
t + -3.5 = 3.041381265

Reorder the terms:
-3.5 + t = 3.041381265

Solving
-3.5 + t = 3.041381265

Solving for variable 't'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + t = 3.041381265 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + t = 3.041381265 + 3.5
t = 3.041381265 + 3.5

Combine like terms: 3.041381265 + 3.5 = 6.541381265
t = 6.541381265

Simplifying
t = 6.541381265

Subproblem 2
t + -3.5 = -3.041381265

Simplifying
t + -3.5 = -3.041381265

Reorder the terms:
-3.5 + t = -3.041381265

Solving
-3.5 + t = -3.041381265

Solving for variable 't'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + t = -3.041381265 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + t = -3.041381265 + 3.5
t = -3.041381265 + 3.5

Combine like terms: -3.041381265 + 3.5 = 0.458618735
t = 0.458618735

Simplifying
t = 0.458618735

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {6.541381265, 0.458618735}
Solution
t = {0, 6.541381265, 0.458618735}