Algebra Summary

Factors

  1. Highest Common Factor† †† Example: 9xy2 - 6x2y2 + 3xy = 3xy(3y - 2xy +1)
  2. Grouping††††††††††††††††††††††††††††††† Example: 2c2 - 4cd - c + 2d = 2c(c - 2d) - 1(c - 2d)
                                                                                                      =(c - 2d)(2c - 1)
  3. Quadratic Factor††††††††††††††††† Example: x2 - 7x + 12 = (x - 4)(x - 3)
  4. Difference of two squares  Formula: x2 - y2 = (x + y)(x - y)
                                                   
    Example: 9x2 - 25y2 = (3x + 5y)(3x - 5y)

Simplify

This instruction can mean:

  1. Add like terms†††††††††††††††††††††††† Example: 3x - 2x2 + 6x + 10x2 - 5x = 4x = 8x2 †††††††††††††††

  2. Expand (multiply) brackets††††Example: (2x - 3)(x + 4) = 2x(x + 4) - 3(x + 4)
                                                                      = 2x2 + 8x -3x -12 = 2x2 + 5x - 12

  3. Divide††††††††† †††††††††††††††††††††††††††  Example: 6x3 - x2 - 33x - 28 ÷ 3x + 4

                                                               


                                                          
  4. Express as a single fraction††††Example:

    1   1 1   1
    x2 - 9   x2 - x - 6 (x - 3)(x + 3)   (x - 3)(x + 4)


    1(x + 2) - 1(x + 3) -1
    (x - 3)(x + 3)(X + 2) (x - 3)(x + 3)(x+ 2)


  5. Cancel††††††††††††††††
    ††††††††††††††††††††    
    Example:

    4x2 - y2 (2x + y)(2x - y) 2x - y
    2xy + y2 y(2x + y) y

Substitution

This means that you should replace certain letters by the numbers you are given.

Example: if x = 9†find the value of

Replacing x by 9:

Manipulating Formula

The instruction given in this case is usually express x in terms of y and z (obviously the letters change). Be very careful that you "do the same thing to both sides of the equation". Remember to "get rid" of something by "doing the opposite to both sides".

Example:      If , express a in terms of b and c.††

Example:      Express a  in terms of b  and c  where   .

Solving Equations

1.        Linear†                Example: ††††††††††††††††††††††††††††††††††††††††
                                   (has xís and numbers)

2.        Fractional linear†Example:
            (has xís and numbers and fractions)

3.        Linear simultaneous††Example:††††††††††††

†                                   (has xís and yís)††††

4.        Quadratics using factors†††† Example:†

Solving Inequalities

1.        Linear inequalities

Treat like equations but multiplying†or dividing by a minus number reverses the inequality sign.

Example:  Solve      

Note

Algebra appears on Paper 1 in Question 1 (usually 40 or 50 marks), Question 3 (50 marks) and Question 6 (20/30 marks)

Thatís a total of about 130 out of 600 marks or 22% just for Algebra!

Read and practice the methods on these sheets and then practice exam questions.