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AQA A-level Maths Manipulating equations

Manipulating equations

This page covers the following topics:

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

Simplifying equations

Expanding means removing brackets by multiplying out. 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.

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 of it together 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 one remember the terms that must be multiplied together to find an expansion of two brackets. 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. Take the sum of the products to find the expansion of the two brackets.

Order of binomial expansions

Factorising is the reverse of expanding. A quadratic expression, xΒ² + cx + d, can be factorised to be written as (x + a)(x + b), where the sum of a and b is equal to c and the product of a and b is equal to d. A special case of factorisation, called the difference between the two squares, is given as: xΒ² βˆ’ yΒ² = (x βˆ’ 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 diagram can be used. The denominators can then be cancelled out and the unkowns can be solved for by equating coefficients on the two sides of the equation.

Partial fractions

The subject of a formula is the variable that is being solved for, and it is found on its own on one side of the equation. Formulas, however, can be rearranged to change the subject of them by performing inverse operations on them.

Changing the subject

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Use the grid method to find the expansion of (6x βˆ’ 7)(3x βˆ’ 2).

AQA A-level Maths Manipulating equations Use the grid method to find the expansion of (6x βˆ’ 7)(3x βˆ’ 2).
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2

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A rectangular book has sides xΒ² + 3 and 4xΒ³ βˆ’ 7. Using the grid method, find the surface area of the book.

AQA A-level Maths Manipulating equations A rectangular book has sides xΒ² + 3 and 4xΒ³ βˆ’ 7. Using the grid method, find the surface area of the book.
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3

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Use the FOIL method to find the expansion of (3xΒ² + 4)(2 βˆ’ x).

AQA A-level Maths Manipulating equations Use the FOIL method to find the expansion of (3xΒ² + 4)(2 βˆ’ x).
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4

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A square has side 4d + 9. Using the FOIL method, find the area of the square.

AQA A-level Maths Manipulating equations A square has side 4d + 9. Using the FOIL method, find the area of the square.
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5

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A block of wood has length of x + 3, width of x βˆ’ 3 and height 5x + 2. Find an expression for the volume of the block using the FOIL method.

AQA A-level Maths Manipulating equations A block of wood has length of x + 3, width of x βˆ’ 3 and height 5x + 2. Find an expression for the volume of the block using the FOIL method.
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6

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Factorise xΒ² βˆ’ 14x βˆ’ 51.

AQA A-level Maths Manipulating equations Factorise xΒ² βˆ’ 14x βˆ’ 51.
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7

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Factorise x⁴ βˆ’ 16.

AQA A-level Maths Manipulating equations Factorise x⁴ βˆ’ 16.
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8

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An earring is shaped and sized as shown in the diagram. Find the perimeter of the earring in its simplest, factorised form.

AQA A-level Maths Manipulating equations An earring is shaped and sized as shown in the diagram. Find the perimeter of the earring in its simplest, factorised form.
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9

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Give (16x + 52)/(2x + 7)Β² as partial fractions.

AQA A-level Maths Manipulating equations Give (16x + 52)/(2x + 7)Β² as partial fractions.
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10

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Give (10xΒ² + 17x + 42)/(x + 10)(xΒ² + 9) as partial fractions.

AQA A-level Maths Manipulating equations Give (10xΒ² + 17x + 42)/(x + 10)(xΒ² + 9) as partial fractions.
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