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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.

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.

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.

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.

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

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.