Expand and simplify 𝑥 minus two multiplied by 𝑥 minus seven.
Before we start this question, it is worth remembering what happens when we multiply positive and negative terms multiplying two positive terms gives us a positive answer. Likewise, multiplying two negative terms gives us a positive answer. Multiplying a positive and a negative term in either order gives us a negative answer.
There are lots of different methods for expanding or multiplying out two brackets. We will look at the FOIL method and the grid method. The FOIL method involves multiplying the first terms, the outside terms, the inside terms, and the last terms.
Multiplying the first terms in each bracket, 𝑥 multiplied by 𝑥, gives us 𝑥 squared. Multiplying the outside terms, 𝑥 multiplied by negative seven, gives us negative seven 𝑥 as a positive multiplied by a negative gives us a negative answer.
Likewise, multiplying the inside terms, negative two and 𝑥, gives us negative two 𝑥. And finally, multiplying the last terms, negative two and negative seven, gives us positive 14. Multiplying two negative numbers gives us a positive answer.
In order to simplify this expression, we need to group or collect the like terms. We need to collect negative seven 𝑥 and negative two 𝑥. Negative seven 𝑥 minus two 𝑥 is equal to negative nine 𝑥. Therefore, 𝑥 minus two multiplied by 𝑥 minus seven is equal to 𝑥 squared minus nine 𝑥 plus 14.
We will now look at an alternative method called the grid method. As with the FOIL method, we need to multiply each of the terms in the first bracket by each of the terms in the second bracket.
𝑥 multiplied by 𝑥 is equal to 𝑥 squared. 𝑥 multiplied by negative seven is equal to negative seven 𝑥. Negative two multiplied by 𝑥 is equal to negative two 𝑥. And finally, negative two multiplied by negative seven is equal to 14 or positive 14.
Once again, we have the same four terms: 𝑥 squared minus seven 𝑥 minus two 𝑥 plus 14. Grouping the 𝑥 terms as we did previously gives us 𝑥 squared minus nine 𝑥 plus 14.
There are lots of other methods for expanding two brackets. As long as you get to the final answer of 𝑥 squared minus nine 𝑥 plus 14, you’ll get the full marks in an exam.