56 



Algebra. 



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11. HiSTOiiiCAii Note. The origin of the solution of the quadratic equa- 

 tion cannot be definitely traced to any one man or any one race. Algebra, as 

 we now have it, has been a slow growth, and, as we pass ba(ik in time, it grad- 

 ually shades off into the arithmetic of antiquity. Biophantus, an Alexandrian 



fv Greek of the fourth century, A. D., who wn te a treatise cm arithmetic, could 

 ] r undoubtedly solve (juadratic equations, although he devotes no special book 

 Vj t^ their treatment. But algebra, in a more perfect form, may be traced to 

 the Hindoos. Aryabhata (475 A. D.) was familiar with a solution of the com- 

 plete quadratic, and Bramagupta (59S A. D.) gives a comparatively elaborate 

 treatment of it. The solution of the quadratic was also known to the Arabs, and 

 a solution with geometric treatment is given by Mohammed ben Musa, of the 

 ninth century. All the early methods of sohition consist in what is commonly 

 known as "conwpleting the square' and w<m(> substantially tlu^ same as those 



12. Examples of Complete Quadkatic Equations, If a 

 quadratic equation cannot be placed in the iorr\^ x'-\-px=q icit/ioiit 

 the introduction of fractions, it is generally advisable before solu- 

 tion to clear it of fractions thereby putting it in the form ax"-^bx 

 =r, in which case a, b, and c will be integral. The equation can 

 then be solved b}^ the method of Art. lo, thereby avoiding frac- 

 tions in the process of solution, which is a great advantage. If 

 the equation takes the form x--{-px=q without p and q being 

 fractional, then a solution by the method of Art. 9 will be better. 



To illustrate the common arrangement of the work we solve the 

 following quadratic. 

 Find the values of x in 



10— -r= 



The first task is to 

 place the eqiiation in 

 one of the typical 

 forms. 



Now "complete the 

 square." 



Since it hi been 

 proved that thi meth- 

 od will give a complete 

 square, it is not nec- 

 essary to work out 

 the value of the co- 

 efficient of X, but mere- 

 ly to indicate it by (). 



■9^+5 = 



3 



Clearing of fractions, 



gx^—2jx-\-i^= 10— A\ 

 Transposing and uniting terms. 

 io.<— 27^1:= — 5. 

 Multiplying througli by 4 times 10, and ad 

 ding (27)' to both vSides, we obtain 



400-r''— (^jX-f (^27/= — 200-|-f27/, 



or 4ooA-^^— ('>"^-27'=529• 



Taking the square root of both members, 



20:l" — 27= ±23, 

 and solving this simple equation 

 20A-=5o or 4 

 x= 2^ or i. 



