3t^2=13t+20

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Solution for 3t^2=13t+20 equation:


Simplifying
3t2 = 13t + 20

Reorder the terms:
3t2 = 20 + 13t

Solving
3t2 = 20 + 13t

Solving for variable 't'.

Reorder the terms:
-20 + -13t + 3t2 = 20 + 13t + -20 + -13t

Reorder the terms:
-20 + -13t + 3t2 = 20 + -20 + 13t + -13t

Combine like terms: 20 + -20 = 0
-20 + -13t + 3t2 = 0 + 13t + -13t
-20 + -13t + 3t2 = 13t + -13t

Combine like terms: 13t + -13t = 0
-20 + -13t + 3t2 = 0

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

Divide each side by '3'.
-6.666666667 + -4.333333333t + t2 = 0

Move the constant term to the right:

Add '6.666666667' to each side of the equation.
-6.666666667 + -4.333333333t + 6.666666667 + t2 = 0 + 6.666666667

Reorder the terms:
-6.666666667 + 6.666666667 + -4.333333333t + t2 = 0 + 6.666666667

Combine like terms: -6.666666667 + 6.666666667 = 0.000000000
0.000000000 + -4.333333333t + t2 = 0 + 6.666666667
-4.333333333t + t2 = 0 + 6.666666667

Combine like terms: 0 + 6.666666667 = 6.666666667
-4.333333333t + t2 = 6.666666667

The t term is -4.333333333t.  Take half its coefficient (-2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
-4.333333333t + 4.694444446 + t2 = 6.666666667 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333t + t2 = 6.666666667 + 4.694444446

Combine like terms: 6.666666667 + 4.694444446 = 11.361111113
4.694444446 + -4.333333333t + t2 = 11.361111113

Factor a perfect square on the left side:
(t + -2.166666667)(t + -2.166666667) = 11.361111113

Calculate the square root of the right side: 3.370624736

Break this problem into two subproblems by setting 
(t + -2.166666667) equal to 3.370624736 and -3.370624736.

Subproblem 1

t + -2.166666667 = 3.370624736 Simplifying t + -2.166666667 = 3.370624736 Reorder the terms: -2.166666667 + t = 3.370624736 Solving -2.166666667 + t = 3.370624736 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.166666667' to each side of the equation. -2.166666667 + 2.166666667 + t = 3.370624736 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + t = 3.370624736 + 2.166666667 t = 3.370624736 + 2.166666667 Combine like terms: 3.370624736 + 2.166666667 = 5.537291403 t = 5.537291403 Simplifying t = 5.537291403

Subproblem 2

t + -2.166666667 = -3.370624736 Simplifying t + -2.166666667 = -3.370624736 Reorder the terms: -2.166666667 + t = -3.370624736 Solving -2.166666667 + t = -3.370624736 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.166666667' to each side of the equation. -2.166666667 + 2.166666667 + t = -3.370624736 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + t = -3.370624736 + 2.166666667 t = -3.370624736 + 2.166666667 Combine like terms: -3.370624736 + 2.166666667 = -1.203958069 t = -1.203958069 Simplifying t = -1.203958069

Solution

The solution to the problem is based on the solutions from the subproblems. t = {5.537291403, -1.203958069}

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