Expanding Equations
Expanding Equations – Introduction
In this article we are going to consider some basic questions regarding expanding equations and to offer solutions and comments as to what the common mistakes are most likely to be from an examiners point of view.
Expanding Equations - Questions 1 - 2
Q1
Solution
y^2+3 y+2 y+6y^2+5 y+6
Comments
The main mistake here is to add rather than multiply the constant terms or to add the powers together. Only add powers when it is multiplication.
Solution
\begin{aligned} & =\frac{3(x-2)}{(x-2)(x-5)} \\ & =\frac{3}{x-5} \end{aligned}Comments
Here it is important to realise that multiplying out the numerator is not going to help. With questions like this it is important to focus on factorising as much as possible. Here the denominator can clearly be factorised and then two terms can be cancelled to give a simplified answer.
Q2.
Solution
\begin{aligned} & 2 x+6+3 x+18 \\ & 5 x+24 \end{aligned}Comments
Questions like these should not cause too many issues as long as you are careful with the actual expansion and collecting like terms. Do be careful if there are any negative signs.
Solution
3 y(y-4)Here there is a number and a letter that is common to both terms. Taking out out the number as the common factor or the letter will not give a complete factorised answer.
Expanding Equations - Questions 3 - 4
Q3.
Solution
8 b cA typical common mistake that could be made here would be of arithmetic.
Solution
6 w-15 tComments
Note here that there are two different variables and so no further simplification can be done. Note also that there is a negative sign in the bracket so care needs to be taken with this multiplication.
Solution
\begin{gathered} x^2-2 x+7 x-14 \\ x^2+5 x-14 \end{gathered}Comments
Quite often the actual expansion can be done but it is the simplification of terms that can lead to errors. Remember that “-2 + 7 = 7 – 2 = 5”. You can always write the numbers the other way round to make the arithmetic more easy.
Q4.
Solution
3 x+6Comments
Remember that you are performing a multiplication so the 3 needs to be multiplied with the 2 also. This step can quite often be missed out or the 3 and the 2 are added together. Note also that there is no further simplification of the answer. 3 x+6 does not equal 9 x.
Solution
6 x y\left(2 x^2-3 y\right)Comments
When you are factorising it is important that you find the highest common factor in order to completely factorise an expression. As well as looking for common numbers you also need to determine if there are any common variables.
Solution
\begin{gathered} 2 x^2+8 x-3 x-12 \\ 2 x^2+5 x-12 \end{gathered}Comments
Here it is important to take care with the different signs and to then collect like terms as required.
Solution
10 x^7 y^5Comments
Here it is a case of adding the powers which is generally done well but then you need to multiply the 5 and the 2 together not add them.
Expanding Equations – Question 5
Q5
Solution
3 x(2 x+3 y)Comments
The main mistake to avoid here is to not find the highest common factor of 6 and 9. Also it is important to look at the variables and to determine any common terms here.
Solution
\begin{aligned} & 2 x^2-4 x+5 x-10 \\ & 2 x^2+x-10 \end{aligned}Comments
The main thing to watch here is the multiplication using the negative sign as well as the collection of like terms.
With questions that involve expanding equations you need to consider and always bear in mind the points that have been mentioned. Knowing what the common mistakes are provides an opportunity to avoid those mistakes yourself and to gain the required marks as needed.