How do I reverse this equation?

Question

I have the following equation but would like to isolate L:

P = 5L^2 + 5L

How do I get L as a function of P?

Note that the "^" symbol represents an exponent.

The Gnostic · Accepted Answer

P = 5L² + 5L

Try quadratic?
5L² + 5L - P = 0
L = (-5 ± √(25 + 20P))/10

Another approach: complete the square:
P/5 = L² + L
(P/5) + 1/4 = L² + L + 1/4
(P/5) + 1/4 = (L + 1/2)²
L + 1/2 = √((P/5) + 1/4)
L = √((P/5 + 1/4) - 1/2

Rapidfire · Answer

Complete the square then you can solve the equation for L:
P = 5L² + 5L
P = 5(L² + L)
4P = 5(4L² + 4L)
4P = 5[(2L + 1)² - 1]
(2L + 1)² - 1 = 4P / 5
(2L + 1)² = 4P / 5 + 1
2L + 1 = √(4P / 5 + 1)
2L = √(4P / 5 + 1) - 1
L = [√(4P / 5 + 1) - 1] / 2
L = [√{(4P + 5) / 5} - 1] / 2
L = [√(4P + 5) / √5 - 1] / 2

How do I reverse this equation?

2 Answers