Algebraic Expressions And Formulae.doc

Crescent Girls’ School

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Algebraic Expressions and Formulae Content 2.1 2.2 2.3 2.4 2.5

Introduction Evaluation of Algebraic Expressions Manipulation of Algebraic Fractions Transformation of Formulae Summary

2.1

Introduction

Some useful common identities: 

By finding common factors E.g. Factorize the expression 6a 3 b  2a 2b  8ab . Each term in the expression contain 2ab . Therefore,



By collecting and regrouping the terms E.g. Factorize 6a  15ab  10b  9a 2



2 2 By using the identity a  b   a  b  a  b  E.g. Factorize 81a 4  16 completely.

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2.2

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By inspection E.g. Find the factors of 2 x 2  7 x  15

Evaluation of Algebraic Expressions

Consider the following example: Example 2a: Evaluate

a2  b2 a 2  b 2  2ab

if a=32 and b=22.

Notice that the working is very tedious if we do not manipulate the algebraic expression before we compute. Thus, a simpler way will be a2  b2 a 2  b 2  2ab

=

See that the computation becomes much simpler and easier to compute. The following Algebraic Rules are very useful in evaluating Algebraic Expression

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Practice 1 Evaluate, without using a calculator, the following expression. 1 1 a 2  2ab  b 2 a) if a  and b   19 20 ab

b2 if a  1.25 and b  11 .25 a

b)

a

c)

1 1 a 2  2ab  b 2 if a  and b   2 2 8 9 2a  ab  b

Consider the following example:

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Example 2b: Evaluate  p  q 2  q given that p  3 and q 

10 . Calculator is allowed. Give your answer 7

correct to 2 decimal places. 

p  q2  q =

Note that: If  p  q 2 is evaluated correct to 2 decimal places before subtracting q , accuracy may be lost.

q = _______________________

Example:  p  q 2 = _________________ 

p  q 2  q = ____________________

which is not the accurate answer. Practice 2:

5 9 , b  5 and c  . 3 4

a)

Evaluate a  b  c 2 given that a 

b)

Evaluate  p  q 2  q given that p  4 and q 

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Consider the following example: Example 2c: Evaluate, without using a calculator, the following expression. ab  cd if a  82 , b  18 , c  41 and d  164 . ab

ab  cd = ab

Observe that 164  82  2 . Thus 82  18  41 2  82 is of the form ab  ac where we can extract the common factor 82. Practice 3 a) Evaluate, without using a calculator, the following expression

ab  cd if a  18 , b  17 , c  6 and d  264 . a2  b2

Homework Practice 2A Pg 26 Q1b; 2c, d; 3b, d; 5a, d, f, i 2.3 Manipulation of Algebraic Fractions We shall revise some of the important skills in manipulating algebraic fractions and consider some harder examples.

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Consider the following example: Example 2d: a) x 2  x

Factorise

b) 18  10 x  8 x 2

a) x 2  x = __________________________ b) 18  10 x  8 x 2 = __________________________

Hence simply

3x 2  3x . 18  10 x  8 x 2

3x 2  3x = 18  10 x  8 x 2

Consider the following example: Example 2e: Simplify

5m  2 p 1  . 2 4m  3 p 4m  mp  3 p 2

5m  2 p 1  = 2 4 m  3p 4m  mp  3 p 2

Practice 4 a)

Simplify

3x  2 3x  1  2 x  3x  2 x  2x 2

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b)

Simplify

3 p  15r  2 p  3r p  9r 2

c)

Simplify

x2 x 3  2 x  3x  2 x  x  6 2

Homework Practice 2B Pg 29 Q3, 4, 5, 6; Q7b, d; Q8q, r, s, t 2.4 Transformation of Formulae Consider the equation a 

x b 3

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It is a __________________ equation. In which x is the unknown and a and b are treated as if they are known numbers. If we replace x by c in this equation, we have a 

c b. 3

We may refer to this equality as a _________________ with ______ as the subject. We can readily find the value of a given values of b and c. However if we are given the value of a and b, where c is the unknown, we can transform the formula by making c the subject before we use substitution. Example: c a   b . Find c when a=5 and b=6. 3 Transform the formula by making c the subject.

Then substitute in the values of a and b.

When solving problems that involve square roots such as the problem below, square both sides of the equation. Consider the following example: Example 2f: a)

If

a  1  2b , make a the subject of the formula.

b)

If t 2 

m2 , make m the subject of the formula. m5

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Practice 5

x v2  1 2 y u

1.

Solve for v in

2.

Make h the subject of the formula

p 1  q 3n

h  2k 3h  k

Homework Practice 2C Pg 32 Q1b, d, e, f, h; Q2c, g, k, m, n Practice 6 1.

Simplify

x 2  3 x  7 18  3 x  x 5 5x

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2.

Given that p 

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qr , make q the subject of the formula. Find the value of q when qr

p  1.2 and r  1450 .

3.

Express s in terms of p and q in 2 ps 2  3qs 2  5 p

4.

Express

2 2x  1 x  2  as a single fraction. x  3 x  4x  3 3  x

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12 x 2  3 when given x  13 . 6x 2  9x  6

5.

Evaluate

2.5

Summary

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Evaluation of Algebraic Expressions We simplify the expression first where possible before computing result. Example: Evaluate if and

Manipulation of Algebraic Fractions Examples: Simplify We factorise the numerator and denominator and then eliminate the common factor. Simplify We factorise the denominator to identify a common denominator. Transformation of Formulae We can transform a formula before we use substitution to solve it. Find x where given that , and .

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