5t(t+1)=20

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Solution for 5t(t+1)=20 equation:


Simplifying
5t(t + 1) = 20

Reorder the terms:
5t(1 + t) = 20
(1 * 5t + t * 5t) = 20
(5t + 5t2) = 20

Solving
5t + 5t2 = 20

Solving for variable 't'.

Reorder the terms:
-20 + 5t + 5t2 = 20 + -20

Combine like terms: 20 + -20 = 0
-20 + 5t + 5t2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-4 + t + t2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-4 + t + t2)' equal to zero and attempt to solve:

Simplifying
-4 + t + t2 = 0

Solving
-4 + t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + t + 4 + t2 = 0 + 4

Reorder the terms:
-4 + 4 + t + t2 = 0 + 4

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

Combine like terms: 0 + 4 = 4
t + t2 = 4

The t term is t.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
t + 0.25 + t2 = 4 + 0.25

Reorder the terms:
0.25 + t + t2 = 4 + 0.25

Combine like terms: 4 + 0.25 = 4.25
0.25 + t + t2 = 4.25

Factor a perfect square on the left side:
(t + 0.5)(t + 0.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(t + 0.5) equal to 2.061552813 and -2.061552813.


Subproblem 1
t + 0.5 = 2.061552813

Simplifying
t + 0.5 = 2.061552813

Reorder the terms:
0.5 + t = 2.061552813

Solving
0.5 + t = 2.061552813

Solving for variable 't'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + t = 2.061552813 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = 2.061552813 + -0.5
t = 2.061552813 + -0.5

Combine like terms: 2.061552813 + -0.5 = 1.561552813
t = 1.561552813

Simplifying
t = 1.561552813


Subproblem 2
t + 0.5 = -2.061552813

Simplifying
t + 0.5 = -2.061552813

Reorder the terms:
0.5 + t = -2.061552813

Solving
0.5 + t = -2.061552813

Solving for variable 't'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + t = -2.061552813 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = -2.061552813 + -0.5
t = -2.061552813 + -0.5

Combine like terms: -2.061552813 + -0.5 = -2.561552813
t = -2.561552813

Simplifying
t = -2.561552813


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {1.561552813, -2.561552813}
Solution
t = {1.561552813, -2.561552813}

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