Quadratic equations – cont’d A
S WE continue to review algebra, I wish to remind you of the following:
I The concepts included in algebra are fairly routine and, with effort, you all should be able to learn them well without much difficulty.
I Many areas, for example, solution of linear equations and in-equations, were done in the lower forms and must be effectively revised prior to the examinations next year.
I Algebra should be selected as one of the compulsory topics in Section 2. I will present, at a later date, the list of topics which are included. We will now review last week’s homework. 1. Solve : 5x + 7 = 3x/2
SOLUTION
The appropriate method is to multiply both sides by 2. (In other examples, with multiple denominators, we find their LCM.) 2 x 5x + 2 x 7 = 2 x 3x/2 10x + 14 = 3x 10x - 3x = 7x = - 14 x = - 14/7 = - 2 2. Solve : x - 2 + x /3 + 1/4 = 1 (Here, there are multiple denominators, so the LCM is used.)
SOLUTION
In this case, the method recommended above may also be used.
As the LCM of 3 and 4 is 12, simplifying the left-hand side: x - 2 /3 + x + 1 /4 4(x - 2) + 3(x + 1)/12 = 4x - 8 + 3x + 3 /12 = 7x - 5 /12 Equating both sides: 7x - 5 /12 = 1 Multiply both sides by 12 7x - 5 = 12 7x = 12 + 5 or 7x = 17. Ans : x = 17/7 3. Factorise: (a) 6x2 - 18x
SOLUTION
As the common factor with respect to 6x2 and 18x is 6x 6x2 - 18x = 6x (x - 3) (b) xy2 + x2y Solution xy2 + x2y = xy(y + x) (xy is the HCF of xy2 and x2y) 4. Factorize: 2mh - 2nh - 3mk + 3nk
SOLUTION
2mh - 2nh - 3mk + 3nk Using grouping, that is, repeated common factor method: 2mh - 2nh - 3mk + 3nk = 2h( m - n) - 3k( m - n)
Please pay careful attention to the negative signs. NB: (m - n) is the common factor (m - n)(2h - 3k) 5. Factorise completely 3x + 6y - x2 - 2xy 3x + 6y - x2 - 2xy = (3x + 6y) - (x2 + 2xy ) Please note the change in the sign within the bracket due to the negative sign in front. It is a step which is often omitted. Factorising each bracket. (3x + 6y) - (x2 + 2xy ) = 3(x + 2y) - x (x + 2y) = (x + 2y) ( 3 - x) Please factorise the following on your own: 1 6mn + 15m - 4n - 10 2 6na - 9ma - 4ny + 6m 3 3t + 6u - t2 - 2tu
EXAMPLES OF METHOD OF FACTORISING QUADRATIC EXPRESSIONS
1. Factorise x2 + 7x + 12 This method is based on the principle that: (x + b)(x + c) = x2 + (b + c) x + bc. Do you see the relationship between (b + c) which is the coefficient of x; bc, which is the constant term; and b and c, which are the values in the brackets on the left-hand side? This relationship, as well as the ‘trial and error’ method, plays an important role in this method.
Given the quadratic expression x2 + (b + c) x + bc, if we determine the values b and c, then the quadratic factors are : (x + b)(x + c).
NB: Given the quadratic expression, the sum of b and c is the coefficient of x, and the product is the constant term. Other methods are also taught. Please practise the one you are comfortable with. Using the above: 1. x2 + 7x + 10 = (x + 5)(x +2) If you have not realised the relationship mentioned above, then please note that: I 5 + 2 = 7 (coefficient of x) I 5 x 2 = 10 (The constant term) You may use ‘trial and error’ to identify 5 and 2, the values which satisfy the relationship. 2. Given, x2 + 6x + 8 as 4 x 2 = 8 and 4 + 2 = 6 Then x2 + 6x + 8 = ( x + 4)( x + 2) 3. Factorise: y2 - 5y + 4. The two numbers whose sum is - 5 and product is 4 are - 4 and - 1. Answer: (y - 4)(y - 1) You may wish to expand the factors to verify your answer. 4. Factorise: 2x2 - 3x - 20 Despite the coefficient of x2 being 2, a method similar to that of example 2 above is used. Trial-and-error method may, therefore, be used. There are other strategies so you need to select the one with which you are comfortable.
2x2 - 3x - 20 = (x - 4)(2x + 5) Please factorise the following on your own: I 3x2 - 5x - 2 I 2x2 + 5x - 12 I 2t2 - 3t - 2
EXAMPLES OF METHOD OF DIFFERENCE OF TWO SQUARES
By expanding (A + B)( A - B) = A2 - B2 This forms the basis of the ‘difference of two squares’ method.
5. Factorise: 9 - 49 x2 This is based on the fact that a2 - b2 = (a - b)(a + b). The critical problem is, therefore, to find the square root of each term. As √9 = 3 and √49 x2 = 7x 9 - 49 x2 = (3 - 7x)(3 + 7x). Answer = (3 - 7x)(3 + 7x). We will try another example. 6. Factorise: 9x2 - 16 By using the method of difference of two squares, you can show that since √9x2 = 3x and √16 = 4, then 9x2 - 16 = (3x - 4)(3x + 4) Answer = (3x - 4)(3x + 4). 7. Factorize: 1 - (x + 2)2 Based on the above, the factors are : {1 + ( x +2)}{1 - ( x + 2)} = (1 + x + 2)(1 - x - 2) Answer: ( 3 + x)( - 1 - x ) Remember to check your answers by expanding the factors and ensure that the product is the same as the given expression. It is important that you review the various methods of factorising a given expression.
ADDITIONAL EXAMPLES
We use the grouping method to factorise the expression 6a +16b + 8ab + 12.
SOLUTION
6a + 16b + 8ab + 12 = 6a + 12 + 8ab + 16b Factorising, = 3(2a + 4) + 4b(2a + 4) = (2a + 4)(3 + 4b) Answer: (2a + 4)(3 + 4b) Factorise the following: 1 - y2 /9
SOLUTION
The square roots are 1 and y/3, the factors are: 1 + y/3 1 - y/3 Please note the following quadratic factors: I Y2 - 2Y - 15 = (Y + 3)(Y - 5) I 3m2 - 10m - 8 = (3m +2)(m - 4)
CONTINUED ON PAGE 24
The Caribbean is a melting pot; as the Jamaican motto so aptly describes it: ‘Out of Many, One People’. We can see ourselves in the Caribbean as “part African, part European, part Asian, part Native American but totally Caribbean”. The Caribbean has emerged as this melting pot of colourful cultures and profound natural beauties.
POSITIVE ATTRIBUTES OF CARIBBEAN CULTURAL DIVERSITY
1 Adds richness to the region. 2. Teaches members to do things differently. 3. Creates strong patriotism. 4. Helps members to learn to appreciate other cultures. 5. Helps to boost the tourism economy. Caribbean countries rely heavily on tourism and so their music, especially the Jamaican reggae, has placed the Caribbean ‘on the map’.
1. State THREE cultural activities that occur/started in your country.
2. Describe TWO ways in which as Caribbean natives we pass on cultural practices.
3. Explain why it is necessary to pass on cultural practices/traditions to the next generation. 4. State THREE reasons for Caribbean cultural diversity. 5. Explain and use correctly terms and concepts associated with social groups and institutions.
Primary group, secondary group, informal group, peer group, voluntary and involuntary membership, culture, institution, social control, folkways, norms, mores and laws.