# StudySquare

This page covers the following topics:

1. Expanding binomials using grids

2. Order of binomial expansions

3. Factorising quadratics

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.

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.

Factorising is the reverse of expanding. A quadratic expression, x² + 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: x² − y² = (x − y)(x + y).

# 1

Multiply out brackets (14 − 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 − x)(8x + 9) = −8x² + 103x + 126.

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# 2

Using the grid method, find the area of a square of side 55 − x². 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 − x²)(55 − x²) = 3025 − 110x² + x⁴.

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# 3

Expand expression (6a³ − 4b)(2a − b) using the FOIL method.

first: 6a³ × 2a = 12a⁴

outer: 6a³ × (−b) = −6a³b

inner: −4b × 2a = −8ab

last: −4b × (−b) = 4b²

(6a³ − 4b)(2a − b) = 12a⁴ − 6a³b − 8ab + 4b².

12a⁴ − 6a³b − 8ab + 4b²

# 4

Use the grid method to find the expansion of (2a − 4ab)(a² + 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 − 4ab)(a² + 4) = −4a³b + 2a³ − 16ab + 8a.

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# 5

Express x² + 27x + 152 in a form of (x + a)(x + b).

The numbers whose sum is 27 and product is 152 are 8 and 19.

x² + 27x + 152 = (x + 8)(x + 19)

(x + 8)(x + 19)

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