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Manipulating equations for OCR GCSE Maths

Manipulating equations

This page covers the following topics:

1. Collecting like terms
2. Basic expansion
3. Expanding binomials using grids
4. Order of binomial expansions
5. Factorising quadratics
6. Changing the subject

One of the ways in which an equation can be simplified is by collecting like terms. This means to simplify the equation by combining together the terms whose variables are the same by adding or subtracting them accordingly. Terms which have the same letter as a factor but to a different power are not considered like terms.

An example of adding like terms is adding the numbers of apples and pears. If a basket has 2 apples and 3 pears, then an apple is eaten and 2 pears are added, it would result in 1 apple and 5 pears being in the basket in the end. Apples and pears separately would be considered the like terms in this example. This could also be written as:
2a + 3p โˆ’ a + 2p =
= 2a โˆ’ a + 3p + 2p =
= a + 5p

Collecting like terms

When there is a single number or term outside a bracket, then the bracket can be expanded by multiplying every term inside the bracket by the one outside it. If the term has a minus symbol in front of it, it should be taken into account when multiplying the numbers inside the brackets.

Basic expansion

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

The subject of a formula is the variable that is being solved for and it is provided on its own on one side of the equation. Formulas can be rearranged to change the subject of them by performing inverse operations on them. For example, changing the subject from y to x in y = b โˆ’ โˆšx would be done by adding โˆšx to both sides and squaring both sides to give x = (b โˆ’ y)ยฒ.

Changing the subject

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

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.

3

A podium is shaped and sized as shown in the diagram. Find a simplified expression for the cross-sectional area of the podium. The area of a rectangle is the multiplication product of its sides.

The total cross-sectional area of the podium can be found by separating it into two rectangles.
area of the top rectangle = xยฒ(x + 1) = xยณ + xยฒ
area of the bottom rectangle = x(3x + 4) = 3xยฒ + 4x
total area of the podium = xยณ + xยฒ + 3xยฒ + 4x = xยณ + 4xยฒ + 4x

xยณ + 4xยฒ + 4x

A podium is shaped and sized as shown in the diagram. Find a simplified expression for the cross-sectional area of the podium. The area of a rectangle is the multiplication product of its sides.

4

Make y the subject of the following formula: xยฒ + 7/y = 8.

Subtracting xยฒ from both sides gives 8 โˆ’ xยฒ = 7/y.
Flipping both sides of the equation gives 1/(8 โˆ’ xยฒ) = y/7.
Multiplying both sides by 7 gives y = 7/(8 โˆ’ xยฒ).

y = 7/(8 โˆ’ xยฒ)

Make y the subject of the following formula: xยฒ + 7/y = 8.

5

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.

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