p^2-42q^2=1

Solution for p^2-42q^2=1 equation:

Simplifying
p2 + -42q2 = 1

Solving
p2 + -42q2 = 1

Solving for variable 'p'.

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

Add '42q2' to each side of the equation.
p2 + -42q2 + 42q2 = 1 + 42q2

Combine like terms: -42q2 + 42q2 = 0
p2 + 0 = 1 + 42q2
p2 = 1 + 42q2

Simplifying
p2 = 1 + 42q2

Reorder the terms:
-1 + p2 + -42q2 = 1 + 42q2 + -1 + -42q2

Reorder the terms:
-1 + p2 + -42q2 = 1 + -1 + 42q2 + -42q2

Combine like terms: 1 + -1 = 0
-1 + p2 + -42q2 = 0 + 42q2 + -42q2
-1 + p2 + -42q2 = 42q2 + -42q2

Combine like terms: 42q2 + -42q2 = 0
-1 + p2 + -42q2 = 0

Factor a trinomial.
(p + -7q)(p + 6q) = 0

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

Simplifying
p + -7q = 0

Solving
p + -7q = 0

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

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

Combine like terms: -7q + 7q = 0
p + 0 = 0 + 7q
p = 0 + 7q
Remove the zero:
p = 7q

Simplifying
p = 7q
Subproblem 2
Set the factor '(p + 6q)' equal to zero and attempt to solve:

Simplifying
p + 6q = 0

Solving
p + 6q = 0

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

Add '-6q' to each side of the equation.
p + 6q + -6q = 0 + -6q

Combine like terms: 6q + -6q = 0
p + 0 = 0 + -6q
p = 0 + -6q
Remove the zero:
p = -6q

Simplifying
p = -6q
Solution
p = {7q, -6q}