6t^2+t-3=0

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Solution for 6t^2+t-3=0 equation:


Simplifying
6t2 + t + -3 = 0

Reorder the terms:
-3 + t + 6t2 = 0

Solving
-3 + t + 6t2 = 0

Solving for variable 't'.

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

Divide each side by '6'.
-0.5 + 0.1666666667t + t2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 0.1666666667t + 0.5 + t2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 0.1666666667t + t2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 0.1666666667t + t2 = 0 + 0.5
0.1666666667t + t2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
0.1666666667t + t2 = 0.5

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.
0.1666666667t + 0.25 + t2 = 0.5 + 0.25

Reorder the terms:
0.25 + 0.1666666667t + t2 = 0.5 + 0.25

Combine like terms: 0.5 + 0.25 = 0.75
0.25 + 0.1666666667t + t2 = 0.75

Factor a perfect square on the left side:
(t + 0.5)(t + 0.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(t + 0.5) equal to 0.866025404 and -0.866025404.

Subproblem 1

t + 0.5 = 0.866025404 Simplifying t + 0.5 = 0.866025404 Reorder the terms: 0.5 + t = 0.866025404 Solving 0.5 + t = 0.866025404 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 = 0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = 0.866025404 + -0.5 t = 0.866025404 + -0.5 Combine like terms: 0.866025404 + -0.5 = 0.366025404 t = 0.366025404 Simplifying t = 0.366025404

Subproblem 2

t + 0.5 = -0.866025404 Simplifying t + 0.5 = -0.866025404 Reorder the terms: 0.5 + t = -0.866025404 Solving 0.5 + t = -0.866025404 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 = -0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + t = -0.866025404 + -0.5 t = -0.866025404 + -0.5 Combine like terms: -0.866025404 + -0.5 = -1.366025404 t = -1.366025404 Simplifying t = -1.366025404

Solution

The solution to the problem is based on the solutions from the subproblems. t = {0.366025404, -1.366025404}

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