Is there an easier way to do it?

[davedave]

Consider the equation x(x-1)(x-2)(x-3)=840

One way to do it is to find factors of 840. By trial and error, 7 which is a factor of 840 satisfies the equation.

Another way is to graph both sides of the equation and find the intersection which is 7.

Is there an easier way to find the answer without using the graphing calculator and the trial and error method?

 

[hamster143]

One way to do it is to find factors of 840.

Only if the equation was helpfully constructed to have integer roots - otherwise factors won't help.

Generally speaking, fourth degree equations are hard to solve. In this particular case, the left hand side has a nice symmetry that allows you to rewrite it as a biquadratic equation: substitute y = x-1.5, express the left hand side as a polynomial, and you get a nice biquadratic equation with roots y = +/- 5.5 and therefore x=7 or x=-4.

 

[BenVitale]

Notice that 29² = 841

If we set B = (x-3)(x-2)(x-1)x

(x-2)(x-1) = x² - 3x + 2

(x-3)x = x² - 3x

(x-2)(x-1) - (x-3)x = 2

A = (x-2)(x-1) - 1 = x² - 3x + 1

A + 1 = (x-2)(x-1)

(x-3)x = A - 1

B = A² - 1

(x-3)(x-2)(x-1)x + 1 = (x² - 3x + 1)² = 29²

We can now solve for x.