Understanding Engineering Mathematics

2.3.11 Properties of quadratic expressions and equations
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A.Factorise the quadratics:

(i) x^2 +x− 2 (ii) x^2 + 6 x+ 9

(iii) x^2 − 81 (iv) 2x^2 + 5 x− 3

(v) 2x^2 − 8 x

B.Solve the quadratic equations obtained by equating the expressions inAto zero.

C.Complete the square:

(i) x^2 + 2 x+ 2 (ii) x^2 − 6 x+ 13

(iii) 4x^2 + 4 x−8(iv)4x^2 − 4 x

D.By completing the square determine the maximum or minimum values (as appropriate)
of the following quadratics, and the values ofxat which they occur.

(i) x^2 + 2 x+ 4 (ii) 16− 4 x− 4 x^2

E.Solve the following equations by both factorisation and formula.

(i) x^2 + 3 x+ 2 = 0 (ii) 2x^2 − 5 x+ 2 = 0

(iii) 3x^2 − 11 x+ 6 =0(iv)x^2 + 10 x+ 16 = 0

(v) x^2 + 2 x− 8 =0(vi)9x^2 + 6 x− 3 = 0

F.Evaluate the following exactly with as little labour as possible and without a calculator.

8 ( 23. 7 )^2 − 10 ( 23. 7 )( 45. 4 )+ 3 ( 45. 4 )^2

4 ( 23. 7 )− 3 ( 45. 4 )

G.For Q2.3.11E confirm your results by calculating the (a) sum and (b) product of the
roots.

2.3.12 Powers and indices for algebraic expressions

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A.Express in the forman

(i) a^2 a^4 (ii) a^3 a^2 a (iii) aa^2 a−^1

(iv) a^7 a^3 /a^2 (v) (a^3 )^2 a−^2 a^3 (vi) a−^21 a^2 (a^3 )^6

(vii) a^5 /a−^3 (viii) (a^2 )^3 /(a^3 )^2

B. Express in the forma^2 n, stating the value ofn.

(i) a^9 a^17 (ii) (a^40 )^1 /^4 (iii) a^3 ((a^3 )^6 a^7 a^5 )^1 /^2

(iv) a^27 a−^3 /a^2