Expand and simplify 𝑥 minus two times 𝑥 plus three.
Let’s consider two different methods: first, the FOIL method; and then we’ll use the grid. The FOIL method gives us order to our expansion, multiplying your firsts then your outers then multiplying your inner terms and then finally multiplying your last terms. Back to our firsts, we multiply 𝑥 times 𝑥; the outers, 𝑥 times three. Remember that we’re adding each of these terms that we multiply together. Inners, negative two times 𝑥. Well it is also true to subtract two times 𝑥 here, keeping the negative with the whole number two prevents sign mistakes later on. Our lasts is multiplying negative two by three.
Our next step is to multiply: 𝑥 times 𝑥 equals 𝑥 squared; 𝑥 times three equals three 𝑥; negative two times 𝑥 equals negative two 𝑥; negative two times three equals negative six. We’re finished FOILing and multiplying, and we’re ready to simplify. We simplify this by combining any like terms. We only have one 𝑥 squared term, so we bring it down. We have two 𝑥 terms, two terms that have 𝑥 to the first power. We can combine them: three 𝑥 plus negative two 𝑥 equals 𝑥. We also have a constant: a negative six. It’s our only constant; there is nothing to combine it with, so we bring it down. And we’ve found the expanded and simplified form to be 𝑥 squared plus 𝑥 minus six.
Let’s look at another method for solving this problem. For the grid method, we’ll take 𝑥 minus two and place 𝑥 in our first box, negative two just below it. We’ll then take 𝑥 plus three and add the 𝑥 to a box in the top row and the positive three to the right of it. And then we’ll multiply. In the second row second column, we would multiply 𝑥 times 𝑥 which equals 𝑥 squared. The next box would be equal to 𝑥 times three which equals three 𝑥. After that we multiply 𝑥 times negative two, which equals negative two 𝑥. And then our final box, negative two times three equals negative six.
And then we combine all four of these terms. 𝑥 squared plus three 𝑥 plus negative two 𝑥 plus negative six. We’ll combine like terms to simplify. There is nothing for us to add to 𝑥 squared. We can combine three 𝑥 and negative two 𝑥, which equals positive 𝑥. There’s nothing to combine the negative six with, so we bring it down. Both methods show the simplified form of 𝑥 minus two times 𝑥 plus three to be 𝑥 squared plus 𝑥 minus six.