ย 

Manipulating equations for Edexcel A-level Maths

Manipulating equations

This page covers the following topics:

1. Expanding binomials using grids
2. Order of binomial expansions
3. Factorising quadratics
4. Partial fractions

One way to expand binomials is by using grids. To do this, assign the terms of the first bracket to the boxes of the grid vertically and assign the terms of the second bracket to the boxes of the grid horizontally. Fill out the grid by multiplying the terms assigned to each box and take the sum of all the products. This will give the expansion of the two brackets.

Expanding binomials using grids

Another way to expand binomials is to use the method of FOIL to determine the order of expansion. FOIL stands for first, outer, inner and last and helps remember the terms that must be multiplied together to find an expansion of two brackets. FOIL method suggests to multiply the two first terms of the brackets, the ones on the outer side of the brackets, the ones on the inner side of the brackets and the two last terms of the brackets. Taking the sum of the products gives the expansion of the two brackets.

Order of binomial expansions

Factorising is the reverse of expanding. A quadratic expression, xยฒ + ax + b, can be factorised to be written as (x + c)(x + d), where the sum of c and d is equal to a and the product of c and d is equal to b. A special case of factorisation, called the difference between the two squares, is given as: xยฒ โˆ’ yยฒ = (x โˆ’ y)(x + y).

Factorising quadratics

Algebraic rational functions can be written as a sum or difference of fractions; this is called partial fractions. To do this, the general forms given in the image can be used. The new fractions can then be added by finding common denominators and coefficients of respective powers of variables equated to get simultaneous equations. An example of splitting a fraction into partial fractions is provided below.

(5x + 7)/(x + 3)(x + 2) = A/(x + 3) + B/(x + 2)
multiplying by (x + 3)(x + 2): 5x + 7 = A(x + 2) + B(x + 3)
expanding: 5x + 7 = Ax + 2A + Bx + 3B
collecting like terms: 5x + 7 = x(A + B) + 2A + 3B
equating coefficients of x: 5 = A + B
equating constants: 7 = 2A + 3B
solving simultaneously: A = 8 and B = โˆ’3
(5x + 7)/(x + 3) (x + 2) = 8/(x + 3) โˆ’ 3(x + 2)

Partial fractions

1

Multiply out brackets (14 โˆ’ x)(8x + 9) by using the grid method.

The terms of the brackets are written on the outsides of the grid.
The products of the terms are found and added together.
Thus, (14 โˆ’ x)(8x + 9) = โˆ’8xยฒ + 103x + 126.

image

Multiply out brackets (14 โˆ’ x)(8x + 9) by using the grid method.

2

Give (10xยฒ + 17x + 42)/(x + 10)(xยฒ + 9) as partial fractions.

(10xยฒ + 17x + 42)/(x + 10)(xยฒ + 9) = A/(x + 10) + (Bx + C)/(xยฒ + 9)
Multiplying both sides by (x + 10)(xยฒ + 9) gives: 10xยฒ + 17x + 42 = A(xยฒ + 9) + (Bx + 9)(x + 10).
Expanding gives: 10xยฒ + 17x + 42 = Axยฒ + 9A + Bxยฒ + 10Bx + Cx + 10C.
Equating coefficients of xยฒ: 10 = A + B.
Equating coefficients of x: 17 = 10B + C.
Equating constants: 42 = 9A + 10C.
Multiplying the first equation by 9: 90 = 9A + 9B.
Subtracting the third equation from this gives: 48 = 9B โˆ’ 10C.
Multiplying the second equation by 10 gives: 170 = 100B + 10C.
Adding this to the previous equation gives: 218 = 109B, so B = 2.
Substituting this in the first equation gives: 10 = A + 2, so A = 8.
Substituting B = 2 in the second equation gives 17 = 20 + C, so C = โˆ’3.
Therefore, (10xยฒ + 17x + 42)/(x + 10)(xยฒ + 9) = 8/(x + 10) + (2x โˆ’ 3)/(xยฒ + 9).

8/(x + 10) + (2x โˆ’ 3)/(xยฒ + 9)

Give (10xยฒ + 17x + 42)/(x + 10)(xยฒ + 9) as partial fractions.

3

Using the grid method, find the area of a square of side 55 โˆ’ xยฒ. The area of a square is a square of one of its sides.

The terms of the brackets are written on the outsides of the grid.
The products of the terms are found and added together.
Then (55 โˆ’ xยฒ)(55 โˆ’ xยฒ) = 3025 โˆ’ 110xยฒ + xโด.

image

Using the grid method, find the area of a square of side 55 โˆ’ xยฒ. The area of a square is a square of one of its sides.

4

Expand expression (6aยณ โˆ’ 4b)(2a โˆ’ b) using the FOIL method.

first: 6aยณ ร— 2a = 12aโด
outer: 6aยณ ร— (โˆ’b) = โˆ’6aยณb
inner: โˆ’4b ร— 2a = โˆ’8ab
last: โˆ’4b ร— (โˆ’b) = 4bยฒ
(6aยณ โˆ’ 4b)(2a โˆ’ b) = 12aโด โˆ’ 6aยณb โˆ’ 8ab + 4bยฒ.

12aโด โˆ’ 6aยณb โˆ’ 8ab + 4bยฒ

Expand expression (6aยณ โˆ’ 4b)(2a โˆ’ b) using the FOIL method.

5

Use the grid method to find the expansion of (2a โˆ’ 4ab)(aยฒ + 4).

The terms of the brackets are written on the outsides of the grid.
The products of the terms are found and added together.
Therefore, (2a โˆ’ 4ab)(aยฒ + 4) = โˆ’4aยณb + 2aยณ โˆ’ 16ab + 8a.

image

Use the grid method to find the expansion of (2a โˆ’ 4ab)(aยฒ + 4).

End of page

ย