5t^2-3t=4t^2+1

Solution for 5t^2-3t=4t^2+1 equation:

Simplifying
5t2 + -3t = 4t2 + 1

Reorder the terms:
-3t + 5t2 = 4t2 + 1

Reorder the terms:
-3t + 5t2 = 1 + 4t2

Solving
-3t + 5t2 = 1 + 4t2

Solving for variable 't'.

Reorder the terms:
-1 + -3t + 5t2 + -4t2 = 1 + 4t2 + -1 + -4t2

Combine like terms: 5t2 + -4t2 = 1t2
-1 + -3t + 1t2 = 1 + 4t2 + -1 + -4t2

Reorder the terms:
-1 + -3t + 1t2 = 1 + -1 + 4t2 + -4t2

Combine like terms: 1 + -1 = 0
-1 + -3t + 1t2 = 0 + 4t2 + -4t2
-1 + -3t + 1t2 = 4t2 + -4t2

Combine like terms: 4t2 + -4t2 = 0
-1 + -3t + 1t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -3t + 1 + t2 = 0 + 1

Reorder the terms:
-1 + 1 + -3t + t2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -3t + t2 = 0 + 1
-3t + t2 = 0 + 1

Combine like terms: 0 + 1 = 1
-3t + t2 = 1

The t term is -3t.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3t + 2.25 + t2 = 1 + 2.25

Reorder the terms:
2.25 + -3t + t2 = 1 + 2.25

Combine like terms: 1 + 2.25 = 3.25
2.25 + -3t + t2 = 3.25

Factor a perfect square on the left side:
(t + -1.5)(t + -1.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(t + -1.5) equal to 1.802775638 and -1.802775638.

Subproblem 1
t + -1.5 = 1.802775638

Simplifying
t + -1.5 = 1.802775638

Reorder the terms:
-1.5 + t = 1.802775638

Solving
-1.5 + t = 1.802775638

Solving for variable 't'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + t = 1.802775638 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + t = 1.802775638 + 1.5
t = 1.802775638 + 1.5

Combine like terms: 1.802775638 + 1.5 = 3.302775638
t = 3.302775638

Simplifying
t = 3.302775638

Subproblem 2
t + -1.5 = -1.802775638

Simplifying
t + -1.5 = -1.802775638

Reorder the terms:
-1.5 + t = -1.802775638

Solving
-1.5 + t = -1.802775638

Solving for variable 't'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + t = -1.802775638 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + t = -1.802775638 + 1.5
t = -1.802775638 + 1.5

Combine like terms: -1.802775638 + 1.5 = -0.302775638
t = -0.302775638

Simplifying
t = -0.302775638

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {3.302775638, -0.302775638}