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GMAT Problems – Quadratic Equations

Quadratic equations are among the most advanced algebra concepts tested on the GMAT. You will encounter quadratic equations on both Problem Solving and Data Sufficiency question types on the GMAT quantitative section. Fear not — they need not be scary! There are just a few basic things you must know.

GMAT Quadratic Equations: The Basics

A quadratic equation is simply a polynomial equation of the second degree. Simply put, it’s an equation that contains a sqaure term. Quadratic equations follow this general form:

ax² + bx + c = 0

where x represents a variable (or unknown), and a, b, and c are constants where a ≠ 0.

Often on the GMAT you will be asked to find the roots of the equation, which is where x = 0. A quadratic equation will either have 2 roots, 1 root, or 0 roots. Check out this simple trick for determining how many real roots a quadratic equation has, which is particularly useful on data sufficiency questions involving quadratic equations.

Generally, the best way to find the roots of a quadratic equation is to factor it. Consider this sample quadratic equation:

x² – x – 12 = 0

What are the factors of this equation?

Answer: (x+3)(x-4) = 0 because if you multiply those factors together using the FOIL method, it produces the original quadratic equation. If either factor = 0, then the entire equation will equal zero. Thus, to find the roots, set each factor equal to zero and solve for x:

If x + 3 = 0, then x = -3.

If x – 4 = 0, then x = 4.

So, the roots of the equation are -3 and 4. In other words, there are two (2) possible values for which the original quadratic equation will equal zero.

On the GMAT

Rather than always trying to factor quadratic equations on the GMAT, you should recognize that the GMAT test makers like to test the following three (3) most common quadratic equations:

  1. x² – y² = (x-y)(x+y)
  2. x² – 2xy + y² = (x-y)²
  3. x² + 2xy + y² = (x+y)²

If you see GMAT sample questions that fall into one of these common quadratic equation patterns, then you should be able to immediately recognize what the factors of the equation are, which will enable you to solve the question much more effectively and efficiently. Try applying that strategy to this sample GMAT question:

Q: If x² – y² = 100, and if x + y = 2, then x – y =

(A) -2
(B) 10
(C) 20
(D) 50
(E) 100

For a step-by-step explanation on how to solve this question, check out this short video:

Here is a full transcript of this video, for your convenience:

“So here’s the questions. If x squared minus y squared equals one hundred, and if x plus
y equals two, then x minus y equals, and you have your answer choices.
So go ahead if you need to go back and look at those common quadratic equations do so.
Give this problem a try and see how you do.
Alright how’d you do? Hopefully you recognize something very very important. Maybe because
I tipped you off to it or maybe because you are starting to recognize this.
And that is you look at a question like this. I’ll use green this time.
If x squared minus y squared equals one hundred and if x plus y equals two then x minus y
equals? Equals what? And we have our answer choices.
And you should have had again those light bulbs going on those red flags going up as
soon as you saw that!
Because you should have recognized that as one of our common quadratic equations. Right?
At the very worst you should have been able to reverse foil it but as soon as you see
a square term light bulbs should be going off and what do we see?
Well we recognize that as that first common quadratic equation. And that first common
quadratic equation, again driving through the fog with headlights on you should do something
you could do something even if don’t see how the whole questions is going to unfold.
We know that x squared minus y squared is one of those common quadratic equations; one
of those three common quadratic equations. The roots of which are what?
What when multiplied together is going to produce that? Well you should remember that
it’s x plus y times x minus y. Right? That’s the whole point. That’s the whole point
in memorizing those and learning those; putting those on flash cards until they become second
nature. Again could you reverse foil that? Could you figure it out on test day? Of course.
But could you also save yourself a lot of time and headache by just learning the common
quadratic equations? Yeah it’s so much easier.
And so then if that equals one hundred, then certainly that has to equal one hundred.
And that’s gonna be in a much more user friendly form.
X plus y times x minus y equals one hundred. Because those things are synonymous.
Those roots produce that and so they must also equal one hundred.
And then we continue to look at the question. So if that equals one hundred which means
those factors multiplied together equals a hundred. And if… x plus y equals two.
Ahhh what do we notice? Well the beauty is we’ve taken that quadratic equation, we’ve
broken it out into its factors and what do we notice?
One of those factors is exactly what the question gives us information about. Right? It’s
no coincidence that the test gives us x plus y equals two. X plus y that term equals two.
So with that times x minus y equals one hundred. And then so what is the question asking? The
question is asking, what then is x minus y. Well x minus y is a term that will be isolated
if we just divide both sides by two. So that were left with x minus y equals fifty.
Right? Now you’re probably just smiling to yourself saying Ahhh that’s interesting
how it all comes together. I couldn’t do it on my own but it looks so easy when you
do it Brett. Well, it should look easy after I do it but now that I’ve done it and now that you have
seen it, hopefully you recognize that this is exactly what you’re looking for. It’s
not going to be any different then this on test day. It just might be different, maybe
it will be a very similar question but for one of the other three common quadratic equations.
In this case we happen to use this one and everything just fell into place and there
ya go. The answer is D. You check it and move on.
Let me say something very very important. Take a look at this. You may notice that in
this question, it is asking us for what is x minus y.
This is something important not only on quadratic equation questions but really all problem
solving questions on the GMAT. Here’s a tip for you.
If you are asked to solve for a term. An entire term. Like in this case x minus y or maybe
you’re solving for two x plus one? Or, what ever it is. It it’s something more than
solving for an individual variable like, what is x. Or what is y? In this case it is x minus
y. You are almost never supposed to solve for the individual variables and then add
them together. Or subtract them together.
So if you started going down the path of what x is and then figuring out what y is and thinking
you could just subtract them to get the right answer, you’re almost never going to be
able to solve these questions that way.
It should be a tip off to you that if you see a question that is asking you to solve
for the term that you want to try to solve for the entire term in tact.
And that should maybe be a little bit maybe driving through the fog with head lights again
it should tell you, hey I need to do something to look for this term in tact.
How can I do that? And that if you were going down the wrong path. It may put you back on
the right path to at least have that realization and maybe you figure some things out about
the quadratic equations just as you saw me do here that isolates the term in its entirety
because you recognize its part of that common quadratic equation.
So just pay attention to that. You want to look for the entire term if it’s asking
you for its entire term rather than breaking it out into its individual component parts.”