Understanding Engineering Mathematics

are factors and we have (x−α)(x−β)≡x^2 +ax+b Expanding the left-hand side gives x^2 −(α+β)x+αβ≡x^2 +ax+b So b=product of roots=α ...

so in this case we get x= − 3 ± √ 32 − 4 × 2 ×(− 2 ) 2 × 2 = − 3 ± √ 25 4 = − 3 ± 5 4 =^12 or − 2 , as above B. To complete the ...

(ab)n=anbn a^0 = 1 wherea,bare algebraic functions,m,nneed not be integers,butcare is needed when they are not. For example √ √ ...

but this soon becomes tedious. Fortunately there is a well established routine method for expanding such expressions. Sincea+bis ...

The expansion for( 1 +x)nis now easily obtained from the above by puttinga=1, b=xto get ( 1 +x)n= 1 +nx+ n(n− 1 ) 2! x^2 + n(n− ...

(v) x^3 − 2 x^2 +x−1(vi)ex+e−x (vii) x^2 − 3 x+ 2 = 0 (viii) 2 √ x−x^2 (ix) 2 x+ 1 (x) sinx−3cosx (xi) s 1 (^3) − 2 s^2 +s= 0 (x ...

B.Expand the following expressions, collecting like terms: (i) (x− 1 )(x+ 2 ) (ii) (x− 1 )(x+ 2 )(x+ 4 ) (iii) (x− 1 )(x+ 1 )^2 ...

2.3.5 Equalities and identities ➤➤ 38 50 ➤ A.Determine the real values ofA,B,C,Din the following identities: (i) (s− 1 )(s+ 2 )≡ ...

(vii) x x^2 − 1 + 1 x− 1 (viii) x− 1 x+ 1 + 2 x+ 2 (ix) 3 x^2 + 1 + 2 x^2 + 2 (x) 2x− 1 + 2 x− 1 − 3 x+ 2 B.Put over a common de ...

2.3.11 Properties of quadratic expressions and equations ➤➤ 39 64 ➤ A.Factorise the quadratics: (i) x^2 +x− 2 (ii) x^2 + 6 x+ 9 ...

C.Reduce to simplest form (i) a^2 b^3 c^2 abc (ii) (a^2 )^3 c^12 ab^2 (iii) ba^12 b^7 c^4 (a^2 b^4 c^6 )^1 /^2 (iv) (a^3 )^4 c^1 ...

whereVis the capacitor voltage. Such equations will be solved in Chapter 15, but for now note that in doing so it is necessary t ...

is described by thedifferential equation dx dt =k(x−a)(x−b)(x−c) wherexis the number ofABCmolecules anda,b,care the initial conc ...

D. (i) 2x+ 4 (ii) 2x+ 1 (iii) 3t^2 − 3 t (iv) s^2 +st− 2 t^2 (v) a^3 − 3 a^2 (vi) x^3 +x^2 − 3 x+ 1 (vii) − 2 u^3 − 6 u (viii) 9 ...

B. (i) (t+ 2 )(t+ 3 ) (ii) (t+ 1 )(t− 2 )(t+ 2 )(See Q2.3.2B(vii)) (iii) (y− 2 )(y− 3 )(y+ 1 )(y+ 2 )(See Q2.3.2B(iv)) (iv) ( 3 ...

2.3.8 Algebra of rational functions A. (i) −x− 8 (x+ 2 )(x− 1 ) (ii) −x− 10 (x+ 3 )(x− 4 ) (iii) −x− 13 (x+ 3 )(x− 2 ) (iv) 5 x− ...

2.3.11 Properties of quadratic expressions and equations A. (i) (x− 1 )(x+ 2 ) (ii) (x+ 3 )^2 (iii) (x− 9 )(x+ 9 ) (iv) (x+ 3 )( ...

(iv) 64− 576 x+ 2160 x^2 − 4320 x^3 + 19440 x^4 − 2916 x^5 + 729 x^6 (v) 243s^5 − 810 s^4 t+ 1080 s^3 t^2 − 720 s^2 t^3 + 240 st ...

3 Functions and Series This chapter gathers together a range of topics relating to functions and their general properties. Some ...

use of inequalities in such topics as linear programming manipulation of functions in other maths topics such as calculus summa ...