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When to add parentheses

Most algebra students realize that they need to respect parentheses in mathematical expressions and equations in order to implement the correct order of operations which starts with parentheses, is followed by exponents, then by multiplication and division, and finally by addition and subtraction. They might also remember this with the acronym PEMDAS. The point of using parentheses is to actually override the default order of operations that would exist without them. For instance, in the expression 3 + 4 * 5, you have to do the multiplication first, getting 3 + 20 or 23; but if we instead have (3 + 4) * 5, you have to do the addition first, getting 7 * 5 or 35. In an expression like (x + 3)*(x + 2), we have to multiply all 4 terms (thus accounting for the FOIL rule). If we did not have parentheses, this expression would be x + 3x + 2 or 4x + 2 which is quite different and does not have a squared term.

However, many students don't realize that they sometimes need to add parentheses to an equation as a result of manipulations they do to it. Failure to do this gives incorrect answers.

Here is an example: We are given the equation x + 2 = y/5 and want to express y in terms of x. So, one wants to multiply both sides of the equation by 5. Some students make the mistake of then writing 5*x + 2 = y and then y = 5x + 2. But this is incorrect! When you multiply each side by 5, you need to multiply the entire left side by 5; to make sure you do this, you should surround x + 2 with parentheses before multiplying by 5. So, you should write 5*(x + 2) = y which results in y = 5x + 10.

Here is another example: We are given the equation x = 2/x-1 (where the / would actually be a horizontal line with x - 1 in the denominator). To solve for x, we need to multiply both sides by x - 1. But when we do this, we need to add parentheses around x - 1, getting (x - 1)*x = 2 which then gives x^2 - x = 2 which gives x^2 - x - 2 = 0 which can then be factored and solved for x (with solutions -1 and 2).

Note that in both cases we had to add parentheses around a sum of terms (where I use "sum" to include addition or subtraction of terms). This is typical. In both cases, we had to add parentheses because we were multiplying two items, one of which included two terms but did not initially have parentheses.

So, remember to add your own parentheses in situations like these to make sure you get the right answers.