GMAT Quadratic Equations
Quadratic equations will come in the form of something that looks like this:
In this example, a, b, and c are constants. In an example that is a bit easier to digest, you will see the equation written out like this:
In the above example, a=1, b=5 and c=6. You will likely not need to know the formula in its abstract form; however, variables may be in place of a constant and you will have to solve for the variable. When that happens it is very important to know that all of the constants are in positive form.
Very rarely will the GMAT have the coefficient for the x2 term be anything other than 1. We will start with the examples where the coefficient is 1 and briefly discuss others in the odd chance that you come across it.
In order to study quadratic equations, let’s first look at an example of two polynomials multiplied together.
Polynomial: an expression consisting of more than one term.
In this case, we will look at (x+2)(x+4). In order to multiply this expression, you need to multiply each term inside each parenthesis by the other terms. While that language sounds a bit convoluted, it can be simplified into a method called “FOIL.”
F: First – the first term inside each parenthesis – in this case x and x
O: Outer- the outer most terms in the expression – in this case x and 4
I: Inner- the inner most terms in the expression – in this case 2 and x
L: Last – the last term inside each parenthesis – in this case 2 and 4
When doing the math you arrive to this expression:
What you will notice when looking back over the work is that we added the coefficients for the x terms, which were 2 and 4, to get 6 and we multiplied them to get 8. Thus, when trying to find a solution, you are looking for two numbers that have a product of the third term of the polynomial (in this case and that add up to the coefficient of the second term of the polynomial (in this case 6).
In other words, we can create two equations:
b+c=6 and bc = 8
I would not recommend writing out the formulas, but rather make sure you are comfortable with the operation and test some of the divisors of the third term in the polynomial, but if you need to write them out for a while, by all means do.
When you have a quadratic equation equal to 0, you will be able to solve for values. Our very first equation was as follows:
If we were to factor the left side of the equation, we would have the following:
Since we know that the only way to get 0 when multiplying is if one or more of the numbers is 0, we can solve for x by setting the expression inside each parenthesis equal to 0.
Thus, the solution to this problem is x = -2 or -3. These values are also called “roots”.
There can be more difficult question types where the equation is not laid out for you from the beginning. Do not get scared, just use the methods you practiced in the section on simplifying expressions and solving equations.
While many of the problems you will have to factor will be unique, there are a couple of examples that you should memorize. Namely these two:
The first is know as the difference between 2 perfect squares. The reason that this works is that the middle terms cancel each other out when you FOIL the equation:
It is a similar reason that gets you a middle term of 2xy in the second example. Instead of the middle terms canceling each other out, they double.
Now that you know the basics. Practice. You must be comfortable with how all of these techniques work. It is the mastery of these skills that will allow you to be creative in finding solutions to the more difficult problems.