6t^2-16t-3=0

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Solution for 6t^2-16t-3=0 equation:


Simplifying
6t2 + -16t + -3 = 0

Reorder the terms:
-3 + -16t + 6t2 = 0

Solving
-3 + -16t + 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 + -2.666666667t + t2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -2.666666667t + 0.5 + t2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -2.666666667t + t2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -2.666666667t + t2 = 0 + 0.5
-2.666666667t + t2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-2.666666667t + t2 = 0.5

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 = 0.5 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667t + t2 = 0.5 + 1.777777780

Combine like terms: 0.5 + 1.777777780 = 2.27777778
1.777777780 + -2.666666667t + t2 = 2.27777778

Factor a perfect square on the left side:
(t + -1.333333334)(t + -1.333333334) = 2.27777778

Calculate the square root of the right side: 1.509230857

Break this problem into two subproblems by setting 
(t + -1.333333334) equal to 1.509230857 and -1.509230857.

Subproblem 1

t + -1.333333334 = 1.509230857 Simplifying t + -1.333333334 = 1.509230857 Reorder the terms: -1.333333334 + t = 1.509230857 Solving -1.333333334 + t = 1.509230857 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 = 1.509230857 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = 1.509230857 + 1.333333334 t = 1.509230857 + 1.333333334 Combine like terms: 1.509230857 + 1.333333334 = 2.842564191 t = 2.842564191 Simplifying t = 2.842564191

Subproblem 2

t + -1.333333334 = -1.509230857 Simplifying t + -1.333333334 = -1.509230857 Reorder the terms: -1.333333334 + t = -1.509230857 Solving -1.333333334 + t = -1.509230857 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 = -1.509230857 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + t = -1.509230857 + 1.333333334 t = -1.509230857 + 1.333333334 Combine like terms: -1.509230857 + 1.333333334 = -0.175897523 t = -0.175897523 Simplifying t = -0.175897523

Solution

The solution to the problem is based on the solutions from the subproblems. t = {2.842564191, -0.175897523}

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