GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

Create An Account

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #43 : Simplifying Algebraic Expressions

Simplify:

$(x+10)(x-10)(x^{2}+100)- (x+5)(x-5)(x^{2}+25)$

Possible Answers:

- 9,375

-5,000

9,375

875

Correct answer:

- 9,375

Explanation:

The difference of squares pattern can be applied twice to each product.

$(x+5)(x-5)(x^{2}+25)$

$= (x^{2}-5^{2})(x^{2}+25)$

$= (x^{2}-25)(x^{2}+25)$

$=\left (x^{2} \right )^{2}-25^{2}$

$= x^{4}-625$

$(x+10)(x-10)(x^{2}+100)$

$= (x^{2}-10^{2})(x^{2}+100)$

$= (x^{2}-100)(x^{2}+100)$

$=\left (x^{2} \right )^{2}-100^{2}$

$= x^{4}-10,000$

$(x+10)(x-10)(x^{2}+100)- (x+5)(x-5)(x^{2}+25)$

$= \left (x^{4}-10,000 \right ) - (x^{4}-625)$

$= x^{4}- x^{4} -10,000 +625$

= - 9,375

Report an Error

Example Question #41 : Simplifying Algebraic Expressions

Simplify.

4x^2+3xy-(x-6y+2)^2

Possible Answers:

3x^2-4x+15xy-24y-36y^2-4

3x^2-4x+15xy+12y-36y^2-4

3x^2-4x+15xy+24y-36y^2-4

5x^2-4x+15xy-24y-36y^2+4

Correct answer:

3x^2-4x+15xy+24y-36y^2-4

Explanation:

4x^2+3xy-(x-6y+2)^2

4x^2+3xy-(x^2-6xy+2x-6xy+36y^2-12y+2x-12y+4)

Simplify what is in the parentheses:

4x^2+3xy-(x^2-12xy+4x+36y^2-24y+4)

Distribute the negative sign outside:

4x^2+3xy-x^2+12xy-4x-36y^2+24y-4

Lastly, combine like terms:

3x^2-4x+15xy+24y-36y^2-4

Report an Error

Example Question #42 : Simplifying Algebraic Expressions

Simplify:

$\frac{3x^3 - 60x^2 - 900x}{3x^2 - 60x}$

Possible Answers:

$\frac{3x}{x-30}$

$\frac{1}{x-30}$

$\frac{x^2 - 20x + 300}{x-30}$

x + 10

Correct answer:

x + 10

Explanation:

The first thing we can do is factor out 3x from both the top and bottom of the expression:

$\frac{3x^3 - 60x^2 - 900x}{3x^2 - 60x} = \frac{3x(x^2 - 20x -300)}{3x(x-30)}$

We can then factor the numerator's polynomial:

$\frac{3x(x-30)(x+10)}{3x(x-30)}$

3x divided by itself and (x-30) divided by itself both cancel to 1, leaving x + 10 as the answer.

Report an Error

Example Question #41 : Simplifying Algebraic Expressions

Simplify the following.

-(2x+4y^2)^2+(3x^2-2y+7y^2+y^4)

Possible Answers:

7x^2+8xy^2+17y^4-2y+7y^2

-x^2-16xy^2-15y^4-2y+7y^2

-x^2-8xy^2-15y^4-2y+7y^2

7x^2+16xy^2+17y^4-2y+7y^2

$7x^{2}+32xy^{2}+17y^{8}-2y+7$

Correct answer:

-x^2-16xy^2-15y^4-2y+7y^2

Explanation:

We can begin by expanding the first parentheses:

-(2x+4y^2)^2+(3x^2-2y+7y^2+y^4)

-(4x^2+8xy^2+8xy^2+16y^4)+(3x^2-2y+7y^2+y^4)

We can now combine the like terms: -(4x^2+16xy^2+16y^4)+(3x^2-2y+7y^2+y^4)

Distribute the negative sign:

-4x^2-16xy^2-16y^4+3x^2-2y+7y^2+y^4

Lastly, combine like terms:

-x^2-16xy^2-15y^4-2y+7y^2

Report an Error

Example Question #1352 : Gmat Quantitative Reasoning

The first two terms of an arithmetic sequence are, in order, and $W^{2}$ . What is the third term?

(Assume is positive.)

Possible Answers:

$W^{2} -W$

$W^{2} +W$

$W^{3}$

$W^{4}$

$2W^{2} -W$

Correct answer:

$2W^{2} -W$

Explanation:

The common difference of an arithmetic sequence such as this is the difference of the second and first terms:

$d = W^{2}- W$

Add this to the second term to obtain the third term:

$W^{2}+ W^{2}- W = 2W^{2}- W$

Report an Error

Example Question #1356 : Problem Solving Questions

The first two terms of an arithmetic sequence are, in order, $\left (W + 3 \right )^{2}$ and $\left (W + 5 \right )^{2}$ . Which of the following is the third term of the sequence?

Possible Answers:

$W^{2}+ 14W + 49$

$W^{2}+ 14W +36$

$2 W^{2}+ 16W + 34$

$W^{2}+ 14W + 41$

$2 W^{2}+ 16W + 41$

Correct answer:

$W^{2}+ 14W + 41$

Explanation:

The terms can be rewritten by squaring each binomial as follows:

The first term is

$\left (W + 3 \right )^{2} =W^{2} + 2 \cdot W \cdot 3 + 3 ^{2} = W^{2} + 6W + 9$

The second term is

$\left (W + 5 \right )^{2} =W^{2} + 2 \cdot W \cdot 5 + 5 ^{2} = W^{2} + 10W + 25$

The common difference of an arithmetic sequence such as this is the difference of the second and first terms:

$\begin{matrix} W^{2} + 10W + 25\\ \underline{W^{2} + \; 6W + \; \; 9} \\\; \; \; \; \; \; \; \; \; \; \; \; 4W+16\end{matrix}$

Add this to the second term to obtain the third:

$\begin{matrix} W^{2} + 10W + 25\\ \underline{ \; \; \; \; \; \; \; \; \; \; \; \; 4W+16} \\W^{2}+ 14W + 41 \end{matrix}$

Report an Error

Example Question #41 : Simplifying Algebraic Expressions

If $x=\sqrt{\frac{25}{100}}$ ,

what is the value of

Possible Answers:

$\frac{2}{5}$

$\frac{1}{2}$

$\frac{1}{5}$

$\frac{5}{10}$

$\frac{1}{4}$

Correct answer:

$\frac{1}{2}$

Explanation:

Simplify.

$x=\sqrt{\frac{25}{100}}=\sqrt{\frac{1}{4}}=\frac{1}{2}$

Report an Error

Example Question #42 : Simplifying Algebraic Expressions

If you were to write $(x-3)^{10}$ in expanded form in descending order of degree, what would the third term be?

Possible Answers:

$810x^{8}$

$405x^{8}$

$45x^{8}$

$243x^{8}$

$945x^{8}$

Correct answer:

$405x^{8}$

Explanation:

By the Binomial Theorem, if you expand $(A+B)^{n}$ , writing the result in standard form, the $r \mathrm{th}$ term (with the terms being numbered from 0 to ) is

$C(n,r)A^{n-r}B^{r}$

Set A=x , B=-3 , n=10 , and r=2 (again, the terms are numbered 0 through , so the third term is numbered 2) to get

$C(10,2)\cdot x^{10-2}\cdot (-3)^{2}$

$=\frac{10!}{8! 2!}\cdot x^{8}\cdot 9$

$=45\cdot x^{8}\cdot 9$

$=405x^{8}$

Report an Error

Example Question #51 : Simplifying Algebraic Expressions

Assume that $x,y \neq 0$ .

Which of the following expressions is equal to the following expression?

$\frac{(x+y)^{2} +(x - y)^{2}}{x^{2}y^{2}}$

Possible Answers:

$\frac{2 }{x^{2}y} + \frac{2 }{xy^{2}}$

$4x^{2}y^{2}$

$\frac{2 }{x^{2}} + \frac{2 }{y^{2}}$

$\frac{2 }{x} + \frac{2 }{y}$

Correct answer:

$\frac{2 }{x^{2}} + \frac{2 }{y^{2}}$

Explanation:

$\frac{(x+y)^{2} +(x - y)^{2}}{x^{2}y^{2}}$

$=\frac{(x^{2}+2xy+y^{2}) + (x^{2}-2xy+y^{2}) }{x^{2}y^{2}}$

$=\frac{x^{2}+x^{2}+2xy-2xy+y^{2}+y^{2} }{x^{2}y^{2}}$

$=\frac{2x^{2}+2y^{2} }{x^{2}y^{2}}$

$=\frac{2x^{2} }{x^{2}y^{2}} + \frac{2y^{2} }{x^{2}y^{2}}$

$=\frac{2 }{y^{2}} + \frac{2 }{x^{2}} = \frac{2 }{x^{2}} + \frac{2 }{y^{2}}$

Report an Error

Example Question #1 : Solving Linear Equations With One Unknown

For what value of $\small N$ would the following equation have no solution?

$\small \small 3 (4x - 7) + 12 = 2 (5x - 3) + N (x - 3)$

Possible Answers:

$\small N= -2$

$\small N= 2$

$\small \small N= 1$

The equation must always have at least one solution regardless of the value of $\small N$ .

$\small \small N= -1$

Correct answer:

$\small N= 2$

Explanation:

Simplify both sides of the equation as much as possible, and solve for $\small x$ in the equation in terms of $\small N$ :

$\small \small \small 3 (4x - 7) + 12 = 2 (5x - 3) + N (x - 3)$

$\small 3 \cdot 4x - 3 \cdot 7 + 12 = 2 \cdot 5x - 2 \cdot 3 + N \cdot x - N \cdot 3$

$\small 12x - 21 + 12 = 10x - 6 + Nx - 3N$

$\small 12x - 9 = (10+N)x + (- 6 - 3N)$

$\small \small 12x - (10+N)x = (- 6 - 3N) + 9$

$\small (2-N)x = 3 - 3N$

$\small x = \frac{3 - 3N}{2 -N}$

$\small x$ has exactly one solution unless the denominator is 0 - that is, $\small N = 2$ . We make sure that this value renders no solution by substituting:

$\small \small \small 3 (4x - 7) + 12 = 2 (5x - 3) + N (x - 3)$

$\small \small \small \small 3 (4x - 7) + 12 = 2 (5x - 3) + 2 (x - 3)$

$\small 12x - 21 + 12 = 10x - 6 + 2x - 6$

$\small 12x - 9 = 12x - 12$

$\small - 9 = - 12$

The equation has no solution, and $\small N = 2$ is the correct answer.

Report an Error

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Computer Science Tutors in San Diego, Biology Tutors in Phoenix, ACT Tutors in San Diego, SSAT Tutors in New York City, French Tutors in Seattle, Physics Tutors in Chicago, Spanish Tutors in Phoenix, SSAT Tutors in Washington DC, Calculus Tutors in San Francisco-Bay Area, English Tutors in San Diego

Popular Courses & Classes

Spanish Courses & Classes in Seattle, ISEE Courses & Classes in Denver, ISEE Courses & Classes in Miami, GRE Courses & Classes in Atlanta, SSAT Courses & Classes in Chicago, GMAT Courses & Classes in Atlanta, GMAT Courses & Classes in Washington DC, MCAT Courses & Classes in Los Angeles, SAT Courses & Classes in Philadelphia, GRE Courses & Classes in San Diego

Popular Test Prep

GRE Test Prep in Atlanta, SSAT Test Prep in New York City, GMAT Test Prep in Atlanta, GMAT Test Prep in Washington DC, ISEE Test Prep in San Francisco-Bay Area, SAT Test Prep in New York City, GRE Test Prep in San Diego, LSAT Test Prep in Miami, GRE Test Prep in Washington DC, ISEE Test Prep in Seattle

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

Email address:
Your name:
Feedback:

GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #43 : Simplifying Algebraic Expressions

Example Question #41 : Simplifying Algebraic Expressions

Example Question #42 : Simplifying Algebraic Expressions

Example Question #41 : Simplifying Algebraic Expressions

Example Question #1352 : Gmat Quantitative Reasoning

Example Question #1356 : Problem Solving Questions

Example Question #41 : Simplifying Algebraic Expressions

Example Question #42 : Simplifying Algebraic Expressions

Example Question #51 : Simplifying Algebraic Expressions

Example Question #1 : Solving Linear Equations With One Unknown

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details