Theory of Quadratics. 67 



If we represent the two roots of (i) by « and b for the sake of 

 brevity, we see from Art. 3, that 



^=-i^+\/p-7and b=-\e—s/\F^f. 

 Substituting these in equation (2), it becomes 

 (x—a)(x—b) = Q>. 



7. Corollary. If all the tef^ms of a quadratic be traiisposed to 

 one side, that member is exactly divisible by x mi7ius a root. 



8. Corollary. The form (x—a)(x—b) = o may be used ititer- 

 changeably with x' -\- ex -\-f= o to represent any quadratic equatio7i. 



9. Theorem. Every quadratic eqtiatiofi with 07ie u?iknown 

 q^iantity has two roots and 07ily tzvo. 



It has been shown that every quadratic equation can be placed 

 in the form 



(x—a)(x—b)=o. 



This equation is satisfied when the left member is zero. But 

 the left member becomes zero when either one of its two factors 

 is zero ; that is, when x=^a or x—b. Because each of these two 

 values of x satisfies the equation it has two roots. But the equa- 

 tion can have no other root ; for if any other value than a or bh^ 

 assigned to x, neither of the factors will be zero, and consequently 

 their product will not be zero. Hence there can be no 77iore than 

 two roots. 



It is not claimed that there must be two different roots. In 

 fact, there is nothing in any of the reasoning thus far which shows 

 that a and b must always have different values. In general, they 

 are different from each other, but a special case would be where 

 they are alike. In this case the quadratic takes the form 



(x—a)(x—a) = o, 

 and we still speak of tzco roots because there are tzco factors and 

 because it is merely a special case of the general truth. To say 

 that an equation has two roots equal to each other is merely an- 

 other way of saying that there is but one value which satisfies the 

 equation. 



