6k^2-4k-9=0

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Solution for 6k^2-4k-9=0 equation:


Simplifying
6k2 + -4k + -9 = 0

Reorder the terms:
-9 + -4k + 6k2 = 0

Solving
-9 + -4k + 6k2 = 0

Solving for variable 'k'.

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

Divide each side by '6'.
-1.5 + -0.6666666667k + k2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -0.6666666667k + 1.5 + k2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -0.6666666667k + k2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -0.6666666667k + k2 = 0 + 1.5
-0.6666666667k + k2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-0.6666666667k + k2 = 1.5

The k term is -0.6666666667k.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667k + 0.1111111112 + k2 = 1.5 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667k + k2 = 1.5 + 0.1111111112

Combine like terms: 1.5 + 0.1111111112 = 1.6111111112
0.1111111112 + -0.6666666667k + k2 = 1.6111111112

Factor a perfect square on the left side:
(k + -0.3333333334)(k + -0.3333333334) = 1.6111111112

Calculate the square root of the right side: 1.269295518

Break this problem into two subproblems by setting 
(k + -0.3333333334) equal to 1.269295518 and -1.269295518.

Subproblem 1

k + -0.3333333334 = 1.269295518 Simplifying k + -0.3333333334 = 1.269295518 Reorder the terms: -0.3333333334 + k = 1.269295518 Solving -0.3333333334 + k = 1.269295518 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + k = 1.269295518 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + k = 1.269295518 + 0.3333333334 k = 1.269295518 + 0.3333333334 Combine like terms: 1.269295518 + 0.3333333334 = 1.6026288514 k = 1.6026288514 Simplifying k = 1.6026288514

Subproblem 2

k + -0.3333333334 = -1.269295518 Simplifying k + -0.3333333334 = -1.269295518 Reorder the terms: -0.3333333334 + k = -1.269295518 Solving -0.3333333334 + k = -1.269295518 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + k = -1.269295518 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + k = -1.269295518 + 0.3333333334 k = -1.269295518 + 0.3333333334 Combine like terms: -1.269295518 + 0.3333333334 = -0.9359621846 k = -0.9359621846 Simplifying k = -0.9359621846

Solution

The solution to the problem is based on the solutions from the subproblems. k = {1.6026288514, -0.9359621846}

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