CHAPTER IV. 



QUADRATIC EQUATIONS CONTAINING ONE UNKNOWN QUANTITY. 



1. DEFINITION. An Equation of the Second Degree, or a 

 Quadratic Equation, is one where the highest degree of any term 

 with reference to the unknown quantities is two. 



It must be remembered that the degree of an equation with 

 reference to any quantity is not spoken of unless the equation is 

 rational and integral with reference to that quantity. See I, 

 Art. 6. 



2. We will consider in this chapter quadratic equations con- 

 taining but one unknown quantity, such as — 



^ 3^+5^=24, (i) 



2x^— |-r=.346, ^ (2) 



^3^i-^-(f-^7)-^=4-v^5> (3) 



m + ^1 x^- + {d— t )x=p -f \^k. (4) 



These equations are all obviously quadratics. But some equa- 

 tions, which are irrational or fractional with reference to x in 

 their present form, drop into the quadratic type as soon as the 

 proper transformations are performed. Thus the equation, 



>/^_x/^ I y/ a—y/ b 



^x+ = —:+----.: (5, a) 



y/ b ^x Va 



may be made integral with reference to x by multiplying through 



by ^ X, the resulting form being 



(V t->/aWx , (^a-s/ bWx 



-^+— ~.~ =1 + y^ 



V b V a 



Transposing and uniting terms, 



{b—a)^x_ 



Transposing the rational parts to the right hand side of the equa- 

 tion, we obtain the form 



{b—a)^x_ __ 

 y/Vb "^ 



