MATHEMATICS. ALGEBRA. 



[SOLUTIONS TO THE EXER3ISEI 



,- 10 + =4 or 1 

 Or. by Rule II.. pB471, 



20 + 36 

 .-. * = == =4, or If. 



6. ' t 170:= 40, or*' jr=210. Completing the 



, , 



1 . !_? 



Extracting the root, r :, - 



8U , 29 

 T = 2 ' 



4 



- .r : 



273 



7. 5j' + 4x = 273. Dividing by 5, 



, 4 

 Completing the square, *' + "5 



2 X13GO ,37 



Extracting the root, z+j = \f ~^ = Ty 



4 _273 4_13G9 

 25 -jH- 35 -- 2 T" 



.-.* = 5-j- = 7, or 7J. 



Or, by Rule II., page 474, 



5x' + 4^ = 273, .'. 10x + 4=\/(273X20 + lG) = 

 V547G = +74 



8. 



= 11. Multiplying by 3, 4x"- x=. 33. 



W 



1 33 



Dividing by 4, x 1 ^-x=-j- 



11 33 

 Completing the square, x j- x + gj = -J- 



1 / 529 , 23 



Extracting the root, x g-= \S -g^-=. -g- 



1 , 23 

 .-. x=^-g=Z,or 2J. 



Or, by Rule II., page 474, 

 4x' x = 33, .-. 8x 1 = ^(33X10+1) = 



529 



529 



1 + 23 



3, or 2J. 



9. 



= 9. Multiplying by x, x 2 + 7x 8 



= 9x, .'. x 2 2z = 8. 

 Completing the square, x 3 2x + 1 = 9. 

 Exlnxcting the root, x 1 + 3, /. x = 4, or 2. 

 =4x+159, .-.5* 1 4x^=156. Dividing by 5, 



10. 



Completing the square, 



15G 



n / 



Extracting tho root, x = = \/ '.,-' 



71 = 

 25 "~ 



28 



Or, l.y Rule II., page 474, 



* _ fa 160, /. lOx 4 - V (150 X 20 + 1C) = 



4 + 50 



tlrntis, 



47 -a 



Jfultiplying by x, r, 17.r = 0. 



.*. C* 3 47x -> Sr>. Dividing by C, 





- 



G 





. -17 



Completing the square, 



'- 4 o'- + (S" 



Extracting the root, 



*-S ! 



x 12-^iS 7 ' or c- 



Or by Rule II., page 474, 



Gx 2 47x = 35, /. 12x 47 = V ( S3 X 21 + 47 1 ) 

 = V 1309 - + 37, .'. x = *f- = 7, or 5. 



i.-Sii*sl-^ 



: 2 



Collecting and transposing, 



13. 48x - . 

 Completing the square, 



fractions, 

 = 14(* 4). 



11. Dividing by 48, 



48' 

 11 



40 



Extracting the root, x ' + ., 



.- or- 



II 



"li" 



X 44 + 



..x = 4, or 1 ,!.. 

 Or, by Rule II., page 474, 



48x 8 + 32x ' = 11, /. 9Gz ' + 32 = 

 32 2 ) = V 313C = + 56, 



_1 32 + 56 i 11 



.'. x - = - = g- = - , or 12 ' 



.-.x = 4, or I, 1 ... 



14 X 4 _ 40.cS ^_ 39 = 0> or ( X 2)2_' 4 o(. c !) no. 



Completing the square, (x 2 ) 2 40 (x) + 400 = M 1 . 

 Extracting the root, x 2 20 = ,J 3G1 = + I !>, 

 .-.X 2 =20 + 19 = lor39, .-.x = + l, or + v '39. 



Completiiig the squiu'e. 



. 1 , 1 . 



Extracting the root, x + 7 a 



a' 4i, 



&= ( 



- 3 



Tlii.s may be regarded as a general formula for the solu- 

 tion of any quadratic equation whatever ; for by putting 

 ular values for a uud l> in this result, wo shall huvn 

 the proper values of x in the quadratic to which 1 

 particular values of the coeificicnts belong. M'o further 

 loam from this formula a few general principles well 

 worthy of notice : thus, 



1. In every quadratic, put into the form x 2 + ax + l> - 

 0, if o s = 46, the two roots, must bo equal; since, in this 



a + 

 case the general form for the roots is x = jT~- ~ ; so 



that each root is half the coefficient of x, taken with 

 changed sign. 



2. If a* bo greater than 46, the two roots are unequal 



