3t(t+1)=2(5t+1)

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Solution for 3t(t+1)=2(5t+1) equation:


Simplifying
3t(t + 1) = 2(5t + 1)

Reorder the terms:
3t(1 + t) = 2(5t + 1)
(1 * 3t + t * 3t) = 2(5t + 1)
(3t + 3t2) = 2(5t + 1)

Reorder the terms:
3t + 3t2 = 2(1 + 5t)
3t + 3t2 = (1 * 2 + 5t * 2)
3t + 3t2 = (2 + 10t)

Solving
3t + 3t2 = 2 + 10t

Solving for variable 't'.

Reorder the terms:
-2 + 3t + -10t + 3t2 = 2 + 10t + -2 + -10t

Combine like terms: 3t + -10t = -7t
-2 + -7t + 3t2 = 2 + 10t + -2 + -10t

Reorder the terms:
-2 + -7t + 3t2 = 2 + -2 + 10t + -10t

Combine like terms: 2 + -2 = 0
-2 + -7t + 3t2 = 0 + 10t + -10t
-2 + -7t + 3t2 = 10t + -10t

Combine like terms: 10t + -10t = 0
-2 + -7t + 3t2 = 0

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

Divide each side by '3'.
-0.6666666667 + -2.333333333t + t2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + -2.333333333t + 0.6666666667 + t2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + -2.333333333t + t2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + -2.333333333t + t2 = 0 + 0.6666666667
-2.333333333t + t2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
-2.333333333t + t2 = 0.6666666667

The t term is -2.333333333t.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333t + 1.361111112 + t2 = 0.6666666667 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333t + t2 = 0.6666666667 + 1.361111112

Combine like terms: 0.6666666667 + 1.361111112 = 2.0277777787
1.361111112 + -2.333333333t + t2 = 2.0277777787

Factor a perfect square on the left side:
(t + -1.166666667)(t + -1.166666667) = 2.0277777787

Calculate the square root of the right side: 1.424000625

Break this problem into two subproblems by setting 
(t + -1.166666667) equal to 1.424000625 and -1.424000625.

Subproblem 1

t + -1.166666667 = 1.424000625 Simplifying t + -1.166666667 = 1.424000625 Reorder the terms: -1.166666667 + t = 1.424000625 Solving -1.166666667 + t = 1.424000625 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + t = 1.424000625 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + t = 1.424000625 + 1.166666667 t = 1.424000625 + 1.166666667 Combine like terms: 1.424000625 + 1.166666667 = 2.590667292 t = 2.590667292 Simplifying t = 2.590667292

Subproblem 2

t + -1.166666667 = -1.424000625 Simplifying t + -1.166666667 = -1.424000625 Reorder the terms: -1.166666667 + t = -1.424000625 Solving -1.166666667 + t = -1.424000625 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + t = -1.424000625 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + t = -1.424000625 + 1.166666667 t = -1.424000625 + 1.166666667 Combine like terms: -1.424000625 + 1.166666667 = -0.257333958 t = -0.257333958 Simplifying t = -0.257333958

Solution

The solution to the problem is based on the solutions from the subproblems. t = {2.590667292, -0.257333958}

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