Essential GMAT Algebra Formulas and Practice Questions

a graph paper with scratch work that could be on gmat algebra -image by magoosh

What are the most essential topics of algebra that you need to know to succeed in GMAT math? How many formulas and equations do you have to memorize? Why must we be tortured by all this math???

The good news is that it doesn’t take a huge amount of time or energy to master the basics. Most likely, you already have the fundamentals down.

magoosh video lesson icon with human figure Of course, it never hurts to review, so you might want to check out these first: Introduction to Algebra and FOIL on the GMAT: Simplifying and Expanding.

 
This post is intended to help you gain the edge if you already have a solid foundation. By recognizing and taking advantage of some common algebraic expressions or patterns, and by remembering rather than memorizing (GMAT Math: Memory vs. Memorizing), you can substantially improve your ability to obtain the correct answer on even the most challenging algebra questions on the GMAT!

Table of Contents

Three Essential Formulas for GMAT Algebra

GMAT Algebra Patterns -- filling in the pieces of a puzzle

Here are two algebraic equations and one formula that you should remember for the GMAT Quant section.

  1. Difference of Two Squares: \(a^2 – b^2 = (a-b)(a+b) \)
  2. Squaring a Binomial: \((a \pm b)^2 = a^2 \pm 2ab + b^2 \)
  3. The Discriminant: \(D = b^2 – 4ac \)

There is a common theme among the three items. In one way or another, they all have something to do with quadratic expressions. These tend to be more difficult to work with than linear equations, simply because the tools look more complicated.

Using the Patterns

The first two items in the list pertain to factoring—you can check out this post for more on that topic: Algebra on the GMAT: How to Factor.

The idea is that you can replace the expression on the left of the equals sign by the one on the right, and vice versa. This technique is often used in simplifying algebraic expressions. The practice problems below will also help you to understand when to use this strategy. The third item, the discriminant, is a powerful tool to help analyze the roots of quadratic equations without ever having to find the roots explicitly.

magoosh video lesson icon with purple figure To see these concepts in action, check out our lesson videos on Factoring – Rational Expressions and Quadratic Equations.

GMAT Algebra Practice Problems

Now let’s see if we can spot and use those three patterns in the two types of questions on GMAT Quant!

Problem Solving Questions

large green square with smaller square removed

  1. ABCD is a square with a side \(y \), and JKLM is a square with side \(x \). If Rectangle S (not shown) with length \(x + y \) has the same area as the shaded region above, what is the width of Rectangle S?

A. \(x \)
B. \(y \)
C. \(y – x \)
D. \(y + x \)
E. \(y^2 – x^2 \)

Click here for the answer

 
C.

First of all, notice that the shaded area, quite literally and visually, is the difference of two squares — Area \(= y^2 – x^2 \). We know from the Difference of Two Squares pattern, this factors into:

Area of ABCD \(= y^2 – x^2 = (y + x)(y – x) \)

Rectangle S is supposed to have the same area, and it has a length of \(y + x \), which is one of the factors in the above expression. Therefore, the other factor \(y – x \) must be the width of Rectangle S.

  1. What is the largest integral value of \(m \) such that the quadratic equation \(x^2 – 10x + m \) will have two unique solutions?

A. \(23 \)
B. \(24 \)
C. \(25 \)
D. \(26 \)
E. \(27 \)

Click here for the answer

 
B.

This question is made for the discriminant! You won’t have to guess and check if you know how to use the formula appropriately. To find \(D \), plug in the following to the formula:

\(a = 1 \)
\(b = -10 \)
\(c = m \)

\(D =\) \(b^2\)\( – 4\)\(a\)\(c\)
\(D =\) \((-10)^2\) \(– 4\)\((1)\)\((m)\) \(= 100 – 4m \)

Now, we know that we’re looking for two real solutions, so we want \(D = 100 – 4m > 0 \).

\(100 – 4m > 0 \implies 100 > 4m \implies 25 > m \)

So, we get \(m < 25 \). Be careful NOT to choose 25 as your answer! The largest integer that is less than 25 would be 24.

Data Sufficiency Questions

(For instructions and tips, see GMAT Data Sufficiency Tips.)

  1. The numbers \(a \), \(b \), and \(c \) are all positive. If \(b^2 + c^2 = 117 \), then what is the value of \(a^2 + c^2 \)?

    (1): \(a – b = 3 \)

    (2): \(\frac{a + b}{a – b} = 7 \)

A. Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.
B. Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are not sufficient.

Click here for the answer

 
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Let \(X = a^2 + c^2 \), the value we are seeking. Now notice that if we subtract the first equation in the prompt from this equation, we get \(a^2 – b^2 = X – 117 \). Thus if we could find the value of \(a^2 – b^2 \), then we could find the value of \(X \).

Statement (1): \(a – b = 3 \)

From this statement alone, we cannot calculate \(a^2 – b^2 \), so we can’t find the value of \(X \). Statement (1), alone and by itself, is insufficient.

Statement (2): \(\frac{a + b}{a – b} = 7 \)

From this statement alone, we cannot calculate \(a^2 – b^2 \), so we can’t find the value of \(X \). Statement (2), alone and by itself, is insufficient.

Statements (1) & (2) combined: Now, if we know both statements are true, then we could multiply these two equations, which cancels the denominator and results in the simple equation \(a + b = 21 \). Then we would have the numerical value of both \(a – b \) and \(a + b \).

Next we can figure out \(a^2 – b^2 \) using the Difference of Two Squares formula: \(a^2 – b^2 = (a-b)(a+b) \).

Finally, we can calculate \(X \) and answer the prompt. Combined, the two statements are sufficient.

  1. Given that \(P + 2Q \) is a positive number, what is the value of \(P + 2Q \)?

    (1): \(Q = 2 \)

    (2): \(P^2 + 4PQ + 4Q^2 = 28 \)

A. Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.
B. Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are not sufficient.

Click here for the answer

 
B. Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

The prompt tells us that \(P + 2Q \) is a positive number, and we want to know the value of \(P \).

Statement (1): \(Q = 2 \)

Obviously, by itself this tells us zilch about \(P \). Alone and by itself, this statement is completely insufficient.

Statement (2): \(P^2 + 4PQ + 4Q^2 = 28 \)

Now, this may be a pattern-recognition stretch for some folks, but this is simply the Squaring a Binomial pattern. It may be clearer if we re-write it like this:

\(P^2 + 2(P)(2Q) + (2Q)^2 = 28 \)

Just think of \(P\) in the role of \(a\) and \(2Q\) in the role of \(b\). Now, applying the pattern, we get:

\(a^2\) \( + 2\) \(a\)\(b\) \( + \) \(b^2\) \( = (\)\(a\)\( + \)\(b\)\()^2\)
\(P^2\)\( + 2\)\((P)\)\((2Q)\)\( + \)\((2Q)^2\)\( = (\)\(P\)\( +\)\( 2Q\)\()^2 = 28\)

With \((P + 2Q)^2 = 28 \), all we have to do is take a square root. Normally, we would have to consider both the positive and the negative square root, but since the prompt guarantees that \(P + 2Q \) is a positive number, we need only consider the positive root, \(P + 2Q = \sqrt{28} \).

This statement allows us to determine the unique value of \(P + 2Q \), so statement (2), alone and by itself, is sufficient.

Cicle with line segment intersecting with one point of the circle and extending slightly beyond the opposite point -image by magoosh

  1. In the diagram above, O is the center of the circle, DC \(= a \) and DO \(= b \). What is the area of the circle?

    (1): \(a^2 – 2ab + b^2 = 36 \)

    (2): \(a + b = 22\)

A. Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.
B. Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are not sufficient.

Click here for the answer

 
A. Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

To find the area of the circle, we need to use Archimedes’ formula, \(A = \pi r^2 \). For that we need the radius, OC, which is not given directly. However, notice that OC = DC – OD, or in other words, \(r = a – b \).

Statement (1): \(a^2 – 2ab + b^2 = 36 \)

A major pattern-matching hit! This is the Squaring a Binomial pattern: \((a – b)^2 = a^2 – 2ab + b^2 \)

\(a^2 – 2ab + b^2 = (a – b)^2 = r^2= 36 \)

In fact, this statement already gives us r^2, so we just have to multiply by \(\pi \) and we have the area. This statement, alone and by itself, is sufficient.

Statement (2): \(a + b = 22 \)

We need \(a – b \), and this statement gives us a value of \(a + b \). If we had more information, perhaps we could use this in combination with other information to find what we want, but since this is all we have, this statement, alone and by itself, is insufficient.

Final Thoughts

GMAT algebra questions do not have to be intimidating. Linear and quadratic expressions and their close relatives make up the bulk of the material that you need to know. By mastering the three essential formulas mentioned above, you will be well on your way to Complete GMAT Domination! (A little too dramatic?)

person holding a flag on the top of a hill representing gmat algebra domination -image by magoosh

Ready to get an awesome GMAT score? Start here.

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10 Responses to Essential GMAT Algebra Formulas and Practice Questions

  1. Hannah November 8, 2018 at 10:43 pm #

    The answer to Question 3 doesn’t make sense to me. Isn’t OC equal to OD since it’s a circle and O is in the center?

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert November 13, 2018 at 5:57 pm #

      Hi Hannah!

      This is a great question. 🙂 Remember – Data Sufficiency diagrams aren’t drawn to scale. Because the length is not explicitly stated, we can’t assume anything about the length of OD even though we have a circle and O is in the center. Therefore, we can’t say that OC is equal to OD. 😀

  2. Chayce December 7, 2017 at 11:17 am #

    I’m probably missing something here, but before I looked up the answer to problem 1, I worked the problem a different way…probably incorrectly, but I’m not sure why yet.

    If we change the first binomial to equal c^2 = 117 – b^2, and then we use that new value to plug into the second given binomial (a^2 + c^2), we get a^2 + (117 – b^2) = 0. So continuing, we move over the “117” to the other side:

    a^2 – b^2 = -117

    which, factored, looks like:

    (a + b)(a – b) = -117

    But even before factoring, it looks like the answer to the question is “-117”, without using either of the prompts; obviously, I must be breaking some rule here…maybe setting the equation equal to “0”.

    Anyhow, assuming that I must move on the prompts, the first one provide helpful info:

    a – b = 3

    We can plug this into our factored equation:

    (a + b)3 = -117 Divide both sides by 3

    a + b = -39 Use prompt one equation to solve for b: (a = 3 + b)

    (3+b) + b = -39 Simplify

    b = -18 Use to Solve for a

    a = -15

    Obviously, these don’t satisfy our given given equations, but could you help me figure out where I’ve gone wrong? I assume it has to do with 1) using the stated info : a, b, c are not negative numbers, and 2) trying to combine the given binomials in the way that I did, setting them equal 0.

    Any help would be most appreciated; and thanks for all y’all do!

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert December 8, 2017 at 11:56 am #

      Hi Chayce,

      You hit the nail on the head here–the issue with your response is the fact that you set the equation equal to “0”. We can’t make assumptions like this on the GMAT. We don’t know what this expression is equal to ‘0’ (which is why we set a^2+c^2 equal to X, an unknown variable, in our explanation for this question). It looks like your other problems-solving instincts are OK in methodology (at least, I don’t see anything else strictly ‘illegal’ in your calculations), though as you pointed out they don’t provide you with an answer to the question. It’s difficult to be able to ‘see’ the path forward in questions like this, which is why it’s useful to really study the answers that we give for difficult DS questions–they can help you to understand how to manipulate these questions in a useful (and legal) way 🙂

  3. Chris December 6, 2015 at 8:44 am #

    Hello,

    In the first practice problem we have b^2 + c^2=17.
    If we combine the two statements we end up with two equations: a-b=3 and a+b=21. Instead of finding the difference of two squares formula we can use the strategy of elimination(or linear combination) for the aforementioned equations. Thus, a=12 and b=9.
    If we substitute b in the initial equation b^2 + c^2=17 => c^2= -64. Nothing on the number line can be squared to yield a negative number. Am i wrong?

    • Phuong October 19, 2016 at 6:32 am #

      I’ve got the same result. c is impossible here since it is stated that a, b, and c are positive integers. So since we are not able to find out c, E should be the correct answer.

      Still, the method makes it seems like we have the right answer.

      • Magoosh Test Prep Expert
        Magoosh Test Prep Expert October 20, 2016 at 8:02 am #

        Chris and Phoung, it looks like you’re both right that c is an impossible value, which would make E the correct answer after all. However, a “hidden trick question” where the final answer is incalculable due to a negative square is not the sort of thing that you’d see on a real GMAT data sufficiency question. So Mike’s proposed strategy for this question is basically right, in the sense that it would work on the GMAT itself. But the format of the question and the answer to the question are indeed wrong. Thanks for bringing this to our attention, Chris and Phuong. I’ll mention this error to our Content Improvement Team, so they can update the post with a better, more correct initial practice problem.

  4. Brittany August 15, 2013 at 4:34 pm #

    Hello,

    Just wondering if there are supposed to be multiple choice answers for all the practice problems? In the solutions section, with the Answer = A, etc, seems to indicate this, but as far as I can see, the only question that has multiple choice answers is #4. Am I missing something here?

    Thanks.

    • Mike MᶜGarry
      Mike August 16, 2013 at 9:51 am #

      Brittany,
      Yes, you are missing something big, in terms of GMAT math. Questions #1-3 are what are called Data Sufficiency questions, which comprise approximately half of the questions on the GMAT Quant section. I’ll recommend these two links to start:
      https://magoosh.com/gmat/2012/introduction-to-gmat-data-sufficiency/
      https://magoosh.com/gmat/2013/gmat-data-sufficiency-tips/
      I don’t know how far along you are in your GMAT prep, but in addition to learning all the math, it is crucial for success on the GMAT to master the Data Sufficiency question format.
      Does this make sense?
      Mike 🙂

      • Daniel September 16, 2015 at 12:43 pm #

        Hello,

        I think the issue is that those people who are practicing the GRE get linked to here and then the answers aren’t provided in the same format which confuses us. For the GRE it’s usually multiple choice or fill in the blank after all.


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