(K+1)(2K+1)=15

Solution for (K+1)(2K+1)=15 equation:

Simplifying
(K + 1)(2K + 1) = 15

Reorder the terms:
(1 + K)(2K + 1) = 15

Reorder the terms:
(1 + K)(1 + 2K) = 15

Multiply (1 + K) * (1 + 2K)
(1(1 + 2K) + K(1 + 2K)) = 15
((1 * 1 + 2K * 1) + K(1 + 2K)) = 15
((1 + 2K) + K(1 + 2K)) = 15
(1 + 2K + (1 * K + 2K * K)) = 15
(1 + 2K + (1K + 2K2)) = 15

Combine like terms: 2K + 1K = 3K
(1 + 3K + 2K2) = 15

Solving
1 + 3K + 2K2 = 15

Solving for variable 'K'.

Reorder the terms:
1 + -15 + 3K + 2K2 = 15 + -15

Combine like terms: 1 + -15 = -14
-14 + 3K + 2K2 = 15 + -15

Combine like terms: 15 + -15 = 0
-14 + 3K + 2K2 = 0

Factor a trinomial.
(-7 + -2K)(2 + -1K) = 0

Subproblem 1
Set the factor '(-7 + -2K)' equal to zero and attempt to solve:

Simplifying
-7 + -2K = 0

Solving
-7 + -2K = 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 + -2K = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -2K = 0 + 7
-2K = 0 + 7

Combine like terms: 0 + 7 = 7
-2K = 7

Divide each side by '-2'.
K = -3.5

Simplifying
K = -3.5
Subproblem 2
Set the factor '(2 + -1K)' equal to zero and attempt to solve:

Simplifying
2 + -1K = 0

Solving
2 + -1K = 0

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

Add '-2' to each side of the equation.
2 + -2 + -1K = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -1K = 0 + -2
-1K = 0 + -2

Combine like terms: 0 + -2 = -2
-1K = -2

Divide each side by '-1'.
K = 2

Simplifying
K = 2
Solution
K = {-3.5, 2}