Solving a super nice quintic equation

Overview

blackpenredpen solves the quintic equation (x−3)^5 + (x−7)^5 = (2x−10)^5 by centering the expressions with the substitution t = x−5. Expanding (t+2)^5 and (t−2)^5 makes the odd-power terms cancel, reducing the equation to t(3t^2+4)(t^2−4)=0 and yielding all five roots: 3, 5, 7, and 5 ± (2√3/3)i.

Key takeaways

Centering the expressions with t = x−5 transforms x−3 and x−7 into the symmetric pair t+2 and t−2.
Adding (t+2)^5 and (t−2)^5 cancels terms with odd powers of 2, leaving only 2t^5, 80t^3, and 160t.
The quintic reduces to 10t(3t^4−8t^2−16) = 0, and the quartic factor becomes quadratic after setting u = t^2.
The factorization produces three real roots, x = 3, 5, and 7, plus the conjugate complex pair x = 5 ± (2√3/3)i.
A degree-five equation has five roots counting multiplicity over the complex numbers; finding only the three real roots does not complete the solution.

Chapters

0:00 Recognizing the Quintic’s Symmetry Around x = 5

The equation is structured so the two left-hand inputs, x−3 and x−7, sum to the right-hand input, 2x−10.
Setting each input to zero gives three real candidate roots: x = 3, x = 7, and x = 5.
Because the equation has degree five, the solution set also includes two complex roots.

2:00 Substitute t = x−5 and Expand the Symmetric Fifth Powers

The substitution t = x−5 rewrites x−3 as t+2, x−7 as t−2, and 2x−10 as 2t.
Using the fifth-power binomial coefficients 1, 5, 10, 10, 5, 1, blackpenredpen expands (t+2)^5 and (t−2)^5.
Adding the expansions cancels the even-in-2 terms, leaving 2t^5 + 80t^3 + 160t = 32t^5.

5:45 Factor the Reduced Equation and Recover All Five Roots

Rearranging and factoring gives 10t(3t^4−8t^2−16) = 0; treating the quartic as quadratic in t^2 yields (3t^2+4)(t^2−4).
The factor t = 0 gives x = 5, while t^2 = 4 gives x = 3 and x = 7.
The factor t^2 = −4/3 gives t = ±(2√3/3)i, hence the complex roots x = 5 ± (2√3/3)i.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, blackpenredpen.