3t^2=t+6

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Solution for 3t^2=t+6 equation:


Simplifying
3t2 = t + 6

Reorder the terms:
3t2 = 6 + t

Solving
3t2 = 6 + t

Solving for variable 't'.

Reorder the terms:
-6 + -1t + 3t2 = 6 + t + -6 + -1t

Reorder the terms:
-6 + -1t + 3t2 = 6 + -6 + t + -1t

Combine like terms: 6 + -6 = 0
-6 + -1t + 3t2 = 0 + t + -1t
-6 + -1t + 3t2 = t + -1t

Combine like terms: t + -1t = 0
-6 + -1t + 3t2 = 0

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

Divide each side by '3'.
-2 + -0.3333333333t + t2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -0.3333333333t + 2 + t2 = 0 + 2

Reorder the terms:
-2 + 2 + -0.3333333333t + t2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -0.3333333333t + t2 = 0 + 2
-0.3333333333t + t2 = 0 + 2

Combine like terms: 0 + 2 = 2
-0.3333333333t + t2 = 2

The t term is -0.3333333333t.  Take half its coefficient (-0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
-0.3333333333t + 0.02777777779 + t2 = 2 + 0.02777777779

Reorder the terms:
0.02777777779 + -0.3333333333t + t2 = 2 + 0.02777777779

Combine like terms: 2 + 0.02777777779 = 2.02777777779
0.02777777779 + -0.3333333333t + t2 = 2.02777777779

Factor a perfect square on the left side:
(t + -0.1666666667)(t + -0.1666666667) = 2.02777777779

Calculate the square root of the right side: 1.424000624

Break this problem into two subproblems by setting 
(t + -0.1666666667) equal to 1.424000624 and -1.424000624.

Subproblem 1

t + -0.1666666667 = 1.424000624 Simplifying t + -0.1666666667 = 1.424000624 Reorder the terms: -0.1666666667 + t = 1.424000624 Solving -0.1666666667 + t = 1.424000624 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + t = 1.424000624 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + t = 1.424000624 + 0.1666666667 t = 1.424000624 + 0.1666666667 Combine like terms: 1.424000624 + 0.1666666667 = 1.5906672907 t = 1.5906672907 Simplifying t = 1.5906672907

Subproblem 2

t + -0.1666666667 = -1.424000624 Simplifying t + -0.1666666667 = -1.424000624 Reorder the terms: -0.1666666667 + t = -1.424000624 Solving -0.1666666667 + t = -1.424000624 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + t = -1.424000624 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + t = -1.424000624 + 0.1666666667 t = -1.424000624 + 0.1666666667 Combine like terms: -1.424000624 + 0.1666666667 = -1.2573339573 t = -1.2573339573 Simplifying t = -1.2573339573

Solution

The solution to the problem is based on the solutions from the subproblems. t = {1.5906672907, -1.2573339573}

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