12k^2+15k=6k+20

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Solution for 12k^2+15k=6k+20 equation:


Simplifying
12k2 + 15k = 6k + 20

Reorder the terms:
15k + 12k2 = 6k + 20

Reorder the terms:
15k + 12k2 = 20 + 6k

Solving
15k + 12k2 = 20 + 6k

Solving for variable 'k'.

Reorder the terms:
-20 + 15k + -6k + 12k2 = 20 + 6k + -20 + -6k

Combine like terms: 15k + -6k = 9k
-20 + 9k + 12k2 = 20 + 6k + -20 + -6k

Reorder the terms:
-20 + 9k + 12k2 = 20 + -20 + 6k + -6k

Combine like terms: 20 + -20 = 0
-20 + 9k + 12k2 = 0 + 6k + -6k
-20 + 9k + 12k2 = 6k + -6k

Combine like terms: 6k + -6k = 0
-20 + 9k + 12k2 = 0

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

Divide each side by '12'.
-1.666666667 + 0.75k + k2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + 0.75k + 1.666666667 + k2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + 0.75k + k2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + 0.75k + k2 = 0 + 1.666666667
0.75k + k2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
0.75k + k2 = 1.666666667

The k term is 0.75k.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75k + 0.140625 + k2 = 1.666666667 + 0.140625

Reorder the terms:
0.140625 + 0.75k + k2 = 1.666666667 + 0.140625

Combine like terms: 1.666666667 + 0.140625 = 1.807291667
0.140625 + 0.75k + k2 = 1.807291667

Factor a perfect square on the left side:
(k + 0.375)(k + 0.375) = 1.807291667

Calculate the square root of the right side: 1.344355484

Break this problem into two subproblems by setting 
(k + 0.375) equal to 1.344355484 and -1.344355484.

Subproblem 1

k + 0.375 = 1.344355484 Simplifying k + 0.375 = 1.344355484 Reorder the terms: 0.375 + k = 1.344355484 Solving 0.375 + k = 1.344355484 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + k = 1.344355484 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + k = 1.344355484 + -0.375 k = 1.344355484 + -0.375 Combine like terms: 1.344355484 + -0.375 = 0.969355484 k = 0.969355484 Simplifying k = 0.969355484

Subproblem 2

k + 0.375 = -1.344355484 Simplifying k + 0.375 = -1.344355484 Reorder the terms: 0.375 + k = -1.344355484 Solving 0.375 + k = -1.344355484 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + k = -1.344355484 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + k = -1.344355484 + -0.375 k = -1.344355484 + -0.375 Combine like terms: -1.344355484 + -0.375 = -1.719355484 k = -1.719355484 Simplifying k = -1.719355484

Solution

The solution to the problem is based on the solutions from the subproblems. k = {0.969355484, -1.719355484}

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