PRACTICAL MATHEMATICS 



CHAPTER I 



1. The resolution of a quadratic expression into factors. 

 Taking the most general form for an expression of the second 

 degree 



b c\ 

 -J 



b 6 2 c b z 



-fc + T-+---T 

 a 4a a 4a 



and the expression can be written as 



( 



= a( x z 

 \ 



in terms of the difference of two squares. 

 Hence the factors will be 



( b 



a ( X+ 2a 



2a J\ 2a 2a J 



The nature of the factors depends upon the form taken by the 

 expression b 2 lac. 



If b z 4>ac is a perfect square, the factors are exact. 



If b 2 4ac is positive and not a perfect square, the expression 

 can be split up into factors, but the numerical parts of each 

 factor can only be given correct to as many significant figures as 

 desired. 



If b 2 4fOc is negative, then the factors can only b given in 

 terms of complex quantities. 



If b 2 4-ac = 0, then the actual expression is itself a perfect 

 square. 



To find the factors of 8x* + I3x - 22 



- 22 = 8(x z + 1-6250 - 2-75) 



= 8{0 2 + 1.6250 + (-8125) 2 - 2-75 - (-8125) 2 } 

 = 8{(0 + 0-8125) 2 - 3-410} 

 = 8{(x + 0-8125) 2 - (1-847) 2 } 

 = 8(x +2-659)(a? -1-035) 



A 



