3t^2+8t-6=0

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Solution for 3t^2+8t-6=0 equation:


Simplifying
3t2 + 8t + -6 = 0

Reorder the terms:
-6 + 8t + 3t2 = 0

Solving
-6 + 8t + 3t2 = 0

Solving for variable 't'.

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

Divide each side by '3'.
-2 + 2.666666667t + t2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 2.666666667t + 2 + t2 = 0 + 2

Reorder the terms:
-2 + 2 + 2.666666667t + t2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 2.666666667t + t2 = 0 + 2
2.666666667t + t2 = 0 + 2

Combine like terms: 0 + 2 = 2
2.666666667t + t2 = 2

The t term is 2.666666667t.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667t + 1.777777780 + t2 = 2 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667t + t2 = 2 + 1.777777780

Combine like terms: 2 + 1.777777780 = 3.77777778
1.777777780 + 2.666666667t + t2 = 3.77777778

Factor a perfect square on the left side:
(t + 1.333333334)(t + 1.333333334) = 3.77777778

Calculate the square root of the right side: 1.943650632

Break this problem into two subproblems by setting 
(t + 1.333333334) equal to 1.943650632 and -1.943650632.

Subproblem 1

t + 1.333333334 = 1.943650632 Simplifying t + 1.333333334 = 1.943650632 Reorder the terms: 1.333333334 + t = 1.943650632 Solving 1.333333334 + t = 1.943650632 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + t = 1.943650632 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + t = 1.943650632 + -1.333333334 t = 1.943650632 + -1.333333334 Combine like terms: 1.943650632 + -1.333333334 = 0.610317298 t = 0.610317298 Simplifying t = 0.610317298

Subproblem 2

t + 1.333333334 = -1.943650632 Simplifying t + 1.333333334 = -1.943650632 Reorder the terms: 1.333333334 + t = -1.943650632 Solving 1.333333334 + t = -1.943650632 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + t = -1.943650632 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + t = -1.943650632 + -1.333333334 t = -1.943650632 + -1.333333334 Combine like terms: -1.943650632 + -1.333333334 = -3.276983966 t = -3.276983966 Simplifying t = -3.276983966

Solution

The solution to the problem is based on the solutions from the subproblems. t = {0.610317298, -3.276983966}

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