A cube of edge length $s > 0$ has the property that its surface area is equal to the sum of its volume and five times its edge length. Compute the sum of all possible values of $s$.
6S^2 =S^3+ 5S, solve for S
6 S^2 = S^3 + 5 S
Subtract S^3 + 5 S from both sides:
-S^3 + 6 S^2 - 5 S = 0
The left hand side factors into a product with four terms:
-S (S - 5) (S - 1) = 0
Multiply both sides by -1:
S (S - 5) (S - 1) = 0
Split into three equations:
S - 5 = 0 or S - 1 = 0 or S = 0
Add 5 to both sides:
S = 5 or S - 1 = 0 or S = 0
Add 1 to both sides:
S = 5 or S = 1 Sum= 1 + 5 = 6
6S^2 =S^3+ 5S, solve for S
6 S^2 = S^3 + 5 S
Subtract S^3 + 5 S from both sides:
-S^3 + 6 S^2 - 5 S = 0
The left hand side factors into a product with four terms:
-S (S - 5) (S - 1) = 0
Multiply both sides by -1:
S (S - 5) (S - 1) = 0
Split into three equations:
S - 5 = 0 or S - 1 = 0 or S = 0
Add 5 to both sides:
S = 5 or S - 1 = 0 or S = 0
Add 1 to both sides:
S = 5 or S = 1 Sum= 1 + 5 = 6
