CUBIC EQUATIONS. 587 



the square root of which is 9 + some decimals ; therefore, rejecting the decimals, 

 try — 9 for the root, which succeeds. For, we find 



x* = - 729 and - 73x = 657 



+ 72 = 72 



729 



To compute the value of R, we have x 2 = 81, and — 4q — 3x 2 = 292 — 243 



= 49. Hence R = a/49 = 7, and y = 5^ = 7 -±l = 8, * = ~ R 2 ~ x = ^= - 



=s 1. Hence the roots are 1, 8, and — 9. I have given this example because the 

 roots 1 and 8 are "approximate," though one is eight times greater than the 

 other. This is, I believe, the extreme limit of their relative magnitudes, which 

 does not occur in any other instance. 



It will be observed also that in this example the approximate roots 1 and 8 

 differ by 7. This is also a kind of limit, and gives rise to the following theorem 

 or corollary: — " Whatever the magnitude of the roots of x 3 + qx — r = 0, whose 

 roots are integers, if two of them differ by 7, or by a smaller number, they can all 

 be found by a process only requiring the extraction of the square root." 



Example 3. Given x z — 1477x + r = 0, to find the roots? Here q — — 1477. 

 I have not deemed it necessary to give the value of r in this and the next examples, 

 for the following reason. The values of the roots usually depend upon q alone : 

 q being given, r has generally only one value, at any rate it has but a paucity of 

 values, and as the solution found must always suit one of those values, I have 

 omitted the consideration of it to avoid the introduction of high numbers, r being 

 generally a much higher number than q. If an arbitrary number were suggested 

 for r, it would only follow that such a proposed equation would not have whole 

 numbers for its roots, and it is such only that we are considering at present. 



Since — q = 1477, ::L ^ = 1969 g- The square root of this is 44 + decimals. 



Try therefore — 44 for the root, and it is found to succeed. 



For, since — 4^ = 5908, subtract 3 . 44 2 = 5808, and there remains 100, which 

 is a square number. Hence z — — 44 and R = \/100 = 10. Therefore x + y 

 — 44, x — y = 10, whence x = 27, y — 17. As a verification, we find that these 

 values give for q or xy + xz + yz or 17 . 27 — 17 . 44 — 27 . 44 the number — 1477. 

 which was given in the proposed equation. 



Example 4. Let us try higher numbers. 



Suppose x 3 — 2089168 x + r = 0. This value of q gives ^^ = 2785557 |. 



The square root of this number is 1668 + decimals, and in fact — 1668 is found 

 to be the root. For we have — iq = 8356672, from which if we subtract 3 times 

 the square of 1668 or 8346672, there remains 10000, which is a square number. 



To find the other roots. Since z = — 1668, and R = V 10000 = 100, 

 i + y= 1668 and x — y = 100. Hence x = 884 and y = 784. 



