Symmetry

YouTube’s algorithm also presented me with equations that could be solved by noticing the symmetry in their form.

Example 1

x + y = 4 x^5 + y^5 = 464

Here, we notice that if x = 2 + u then y = 2 - u and:

(2 + u)^5 + (2 - u)^5 = 464

The odd powers of u will cancel out on the expansion of the term on the left, and the even powers will appear twice, to give:

2(2^5 + 10 \times 2^3u^2 + 5 \times 2u^4) = 464 64 + 160u^2 + 20u^4 = 464 u^4 + 8u^2 - 20 = 0

We notice that -20 = -2 \times 10 and 8 = -2 + 10, so we can factor:

(u^2 - 2)(u^2 + 10) = 0

and

u^2 = 2, \quad u^2 = -10

So, we have four solutions:

u = \pm \sqrt{2}, \quad u = \pm i \sqrt{10}

and so:

x=2 \pm \sqrt{2}, \quad y = 2 \mp \sqrt{2} x=2 \pm i\sqrt{10}, \quad y = 2 \mp i\sqrt{10}

Example 2

a^2 - b = 241, \quad b^2-a = 241, \quad a \ne b

So:

a^2 - b = b^2 - a a^2 - b^2 - b + a = 0 (a + b)(a - b) + (a - b) = 0 (a + b + 1)(a - b) = 0

So:

a + b = -1

If a = -1/2 + u then b = -1/2 - u and:

(-1/2 + u)^2 - (-1/2 - u) = 241 (u^2 - u + 1/4) + 1/2 +u = 241 u^2 = 240 + 1/4 = 961/4 = 31^2/4

So, u = \pm31/2 and a = 15, b = -16 (or a = -16, b = 15).

Alternatively:

a^2 + (a + 1) = 241 a^2 + a - 240 = 0

We notice that -240 = -15 \times 16 and 1 = -15 + 16, so we can factor:

(a - 15)(a + 16) = 0

with the same solutions as above.