Fun algebra problem

Number Theory Level 3

Solve the equation

$(x-5)(x-7)(x+6)(x+4)=504$

What is the biggest positive integer root $x$ of the equation?

$\textbf{(This is not an original problem.)}$

3 solutions

Sunil Pradhan
Mar 3, 2015

(x - 5)(x - 7)(x + 6)(x + 4) = 504

(x - 5)(x + 4)(x + 6)(x – 7) = 504

(x² - x – 20)(x² – x – 42) = 504 Let (x² – x) = a

(a – 20)(a – 42) = 504

a² – 62a + 840 – 504 = 0

(a - 56) (a - 6) = 0

Greatest value a = 56 = x² – x = x(x – 1)

7 × 8 = 56

x = 8

William Isoroku
Mar 30, 2015

Lol, just graph it!

Donglin Loo
Dec 13, 2014

Substitutea=x-5 a(a-2)(a+11)(a+9)=504 a(a+9)×(a-2)(a+11)=504 (a^2+9a)×(a^2+9a-22)=504

Substitute t =a^2+9a again t(t-22)=504 t(t-22)=36×22 t=36

a^2+9a=36 a^2+9a-36=0 (a+12)(a-3)=0 a=-12 or a=3 x-5=-12 ,x-5=3 x=-7, x=8 8 is bigger, which is the answer we need.