32 A new Quadratic Theorem. [Jan. 



term, and bring down the two next terms to the remainder for a 

 dividend. 



3. Divide the dividend by double the root, and set the result 

 both in the quotient and divisor. 



4. Multiply the divisor thus increased by the sum last put in 

 the quotient, and subtract the product from the dividend, and 

 so on as in common arithmetic. 



According to this ride, let us now extract the square root 

 from the compound quantity : 



4 a 4 + 12 a 3 x + 13 a-x" + 6a x 3 + x* (2 a 2 + 3 a x + x* sq. root 

 4 a* 



4a' i + 3ax)l2a 3 x + \3a\v 2 

 12a 3 x+ 9a"x- 



4a* + 6a x + x°-) 4a*x°-+6ax 3 + .i* 

 4a°-x- + 6ax 3 +x* 







Let us now work this sum according to the improved theorem, 

 and mark the difference : 



Rule. 



1. Arrange the compound quantity according to the dimen- 

 sions of some letter, and set the root of the first term in the 

 quotient underneath. 



2. Multiply the root by 2, and divide the second term by the 

 product, placing the quotient under the second term. 



3. Multiply the last quotient by 2, and divide the third term 

 by the product, placing the quotient under the third term. 



4. Square the last quotient, and by the product divide the 

 last term ; if nothing remain, the square number is measured by 

 the square root thus found. Like signs give + , unlike signs — . 



This rule answers for quadrinomials, pentanomials, and hexa- 

 nomials ; for by cutting off the mediate terms, as may appear 

 from the following examples, pentanomials and hexanomials are 

 reduced to quadrinomials. In trinomials the second or last term 

 must be squared. According to this rule, extract the square root 

 from 



4 a 4 + 12 a 3 x 

 2 a 9 + 3 a x 



+ 13 a 2 * 2 I + 6 a x 3 + x* 



J + a 2 + square root. 



Here the root of the first term 4 a 4 is 2 a", which we place in 

 the quotient under the first term : secondly, we multiply this 

 root by 2, giving for a product + 4 a\ by which we divide the 

 second term + 12 a 3 x. The quotient + 301" is placed under 

 the second term. We now pass over the mediate term + 13 a- j 2 



