Quadratic Equations. 55 



9. Problem. To Solve the Typical Quadratic a -{■px=q. 

 Add ^p"" to both members and we obtain 



The left hand member is seen on inspection to be the square of 

 the binomial (x-\-'^P) 5 whence taking the square root of both 

 members, 



X-^\P=^s/qJ^^^p-. 

 Solving this simple equation for x we have 



w4iich gives the two values of a , 



-i/+ ^^+iA-' and -i/- ^/c/+i/>^ 

 Hence, to solve an equafion in the fortn x^-{-px=q, add the square 

 of one-half the coefficiejit of x to each side of the equation. Take the 

 square root of both members, and an equation of the first decree is 

 obtaified, from ivhich x can be found in the usual ivay. 



10. Problem. To Solve the Typical Quadratic ax^-\-bx 



Multiply through b}^ ^a and obtain 



^cl\x^ -f- \abx= \ac. 



Adding b" to both members it becomes 



4a^r^ + d^abx -f- b-=^ac-\- b\ 



The left hand member is seen on inspection to be the square of a 



binomial ; whence, taking the square root of both members, we 



obtain 



2ax-\-b=^±^ ^ac-\- b\ 



whence, solving this simple equation, 



— b±i^'\ac^t 

 ,r= - 



2a 



which gives the two values of .v, 



— b-\-\^±ac-\-b~ J —b—^/^ac-\-b' 



and 



2a 2a 



Hence, to solve an equation in the form ax'-\-bx=c, multiply 



through by four times the coefficient of -t" and add the square of the 



coefficient of X to each side of the equation. Then take the square 



root of both members, a?id an equation of the first degree will be ob- 



t allied, fro))i ivhich x can be found. 



