2K(K+3)=9

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Solution for 2K(K+3)=9 equation:


Simplifying
2K(K + 3) = 9

Reorder the terms:
2K(3 + K) = 9
(3 * 2K + K * 2K) = 9
(6K + 2K2) = 9

Solving
6K + 2K2 = 9

Solving for variable 'K'.

Reorder the terms:
-9 + 6K + 2K2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 6K + 2K2 = 0

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

Divide each side by '2'.
-4.5 + 3K + K2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + 3K + 4.5 + K2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + 3K + K2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + 3K + K2 = 0 + 4.5
3K + K2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
3K + K2 = 4.5

The K term is 3K.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3K + 2.25 + K2 = 4.5 + 2.25

Reorder the terms:
2.25 + 3K + K2 = 4.5 + 2.25

Combine like terms: 4.5 + 2.25 = 6.75
2.25 + 3K + K2 = 6.75

Factor a perfect square on the left side:
(K + 1.5)(K + 1.5) = 6.75

Calculate the square root of the right side: 2.598076211

Break this problem into two subproblems by setting 
(K + 1.5) equal to 2.598076211 and -2.598076211.

Subproblem 1

K + 1.5 = 2.598076211 Simplifying K + 1.5 = 2.598076211 Reorder the terms: 1.5 + K = 2.598076211 Solving 1.5 + K = 2.598076211 Solving for variable 'K'. Move all terms containing K to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + K = 2.598076211 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + K = 2.598076211 + -1.5 K = 2.598076211 + -1.5 Combine like terms: 2.598076211 + -1.5 = 1.098076211 K = 1.098076211 Simplifying K = 1.098076211

Subproblem 2

K + 1.5 = -2.598076211 Simplifying K + 1.5 = -2.598076211 Reorder the terms: 1.5 + K = -2.598076211 Solving 1.5 + K = -2.598076211 Solving for variable 'K'. Move all terms containing K to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + K = -2.598076211 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + K = -2.598076211 + -1.5 K = -2.598076211 + -1.5 Combine like terms: -2.598076211 + -1.5 = -4.098076211 K = -4.098076211 Simplifying K = -4.098076211

Solution

The solution to the problem is based on the solutions from the subproblems. K = {1.098076211, -4.098076211}

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