5k(k+1)+5=6(k^2+9)+19k

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Solution for 5k(k+1)+5=6(k^2+9)+19k equation: 


Simplifying
5k(k + 1) + 5 = 6(k2 + 9) + 19k

Reorder the terms:
5k(1 + k) + 5 = 6(k2 + 9) + 19k
(1 * 5k + k * 5k) + 5 = 6(k2 + 9) + 19k
(5k + 5k2) + 5 = 6(k2 + 9) + 19k

Reorder the terms:
5 + 5k + 5k2 = 6(k2 + 9) + 19k

Reorder the terms:
5 + 5k + 5k2 = 6(9 + k2) + 19k
5 + 5k + 5k2 = (9 * 6 + k2 * 6) + 19k
5 + 5k + 5k2 = (54 + 6k2) + 19k

Reorder the terms:
5 + 5k + 5k2 = 54 + 19k + 6k2

Solving
5 + 5k + 5k2 = 54 + 19k + 6k2

Solving for variable 'k'.

Reorder the terms:
5 + -54 + 5k + -19k + 5k2 + -6k2 = 54 + 19k + 6k2 + -54 + -19k + -6k2

Combine like terms: 5 + -54 = -49
-49 + 5k + -19k + 5k2 + -6k2 = 54 + 19k + 6k2 + -54 + -19k + -6k2

Combine like terms: 5k + -19k = -14k
-49 + -14k + 5k2 + -6k2 = 54 + 19k + 6k2 + -54 + -19k + -6k2

Combine like terms: 5k2 + -6k2 = -1k2
-49 + -14k + -1k2 = 54 + 19k + 6k2 + -54 + -19k + -6k2

Reorder the terms:
-49 + -14k + -1k2 = 54 + -54 + 19k + -19k + 6k2 + -6k2

Combine like terms: 54 + -54 = 0
-49 + -14k + -1k2 = 0 + 19k + -19k + 6k2 + -6k2
-49 + -14k + -1k2 = 19k + -19k + 6k2 + -6k2

Combine like terms: 19k + -19k = 0
-49 + -14k + -1k2 = 0 + 6k2 + -6k2
-49 + -14k + -1k2 = 6k2 + -6k2

Combine like terms: 6k2 + -6k2 = 0
-49 + -14k + -1k2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(49 + 14k + k2) = 0

Factor a trinomial.
-1((7 + k)(7 + k)) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(7 + k)' equal to zero and attempt to solve:

Simplifying
7 + k = 0

Solving
7 + k = 0

Move all terms containing k to the left, all other terms to the right.

Add '-7' to each side of the equation.
7 + -7 + k = 0 + -7

Combine like terms: 7 + -7 = 0
0 + k = 0 + -7
k = 0 + -7

Combine like terms: 0 + -7 = -7
k = -7

Simplifying
k = -7

Subproblem 2
Set the factor '(7 + k)' equal to zero and attempt to solve:

Simplifying
7 + k = 0

Solving
7 + k = 0

Move all terms containing k to the left, all other terms to the right.

Add '-7' to each side of the equation.
7 + -7 + k = 0 + -7

Combine like terms: 7 + -7 = 0
0 + k = 0 + -7
k = 0 + -7

Combine like terms: 0 + -7 = -7
k = -7

Simplifying
k = -7
Solution
k = {-7, -7}

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