GMAT Math : Solving Quadratic Equations

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 #1 : Dsq: Solving Quadratic Equations

Consider the equation $x^{2} +Bx = C$

How many real solutions does this equation have?

Statement 1: There exists two different real numbers m, n such that m+n=B and mn=-C

Statement 2: is a positive integer.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

$x^{2} +Bx = C$ can be rewritten as $x^{2} +Bx - C = 0$

If Statement 1 holds, then the equation can be rewritten as (x+m) (x+n)=0 . This equation has solution set $\left \{ -m,-n\right \}$ , which comprises two real numbers.

If Statement 2 holds, the discriminant $B^{2} -4 \cdot1\cdot(-C)= B^{2} +4C$ is positive, being the sum of a nonnegative number and a positive number; this makes the solution set one with two real numbers.

Report an Error

Example Question #2 : Solving Quadratic Equations

Let B,C be two positive integers. How many real solutions does the equation $x^{2} + Bx + C = 0$ have?

Statement 1: is a perfect square of an integer.

Statement 2: $C = \left ( \frac{B}{2} \right ) ^{2}$

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

The number of real solutions of the equation $Ax^{2} + Bx + C = 0$ depends on whether discriminant $B^{2} - 4A C$ is positive, zero, or negative; since A = 1 , this becomes $B^{2} - 4C$ .

If we only know that is a perfect square, then we still need to know to find the number of real solutions. For example, let C = 1 , a perfect square. Then the discriminant is $B^{2} - 4$ , which can be positive, zero, or negative depending on .

But if we know $C = \left ( \frac{B}{2} \right ) ^{2}$ , then the discriminant is

$B^{2} - 4C = B^{2} - 4 \cdot \left ( \frac{B}{2} \right ) ^{2} = B^{2} - 4 \cdot \frac{B^{2}}{4} = B^{2} - B^{2} = 0$

Therefore, $x^{2} + Bx + C = 0$ has one real solution.

Report an Error

Example Question #154 : Algebra

Does the solution set of the following quadratic equation comprise two real solutions, one real solution, or one imaginary solution?

$Ax^{2} -3Ax + B = 0$

Statement 1: B = 2A

Statement 2: B = A + 2

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

The sign of the discriminant of the quadratic expression answers this question; here, the discriminant is

$(-3A)^{2} - 4AB$ ,

$9A^{2} - 4AB$

If we assume Statement 1 alone, this expression becomes

$9A^{2} - 4AB = 9A^{2} - 4A (2A) = 9A^{2} - 8A^{2} = A^{2}$

Since we can assume is nonzero, $A^{2} > 0$ . This makes the discriminant positive, proving that there are two real solutions.

If we assume Statement 2 alone, this expression becomes

$9A^{2} - 4AB = 9A^{2} - 4A \left ( A + 2\right ) = 9A^{2} - 4A^{2}-8A = 5A^{2}-8A$

The sign of $5A^{2}-8A$ can vary.

Case 1: A = 1

Then $5A^{2}-8A = 5 \cdot 1^{2}-8\cdot 1 = 5 - 8 = -3 < 0$

giving the equation two imaginary solutions.

Case 2: A = 2

Then $5A^{2}-8A = 5 \cdot 2^{2}-8\cdot 2 = 20 - 16 = 4 > 0$

giving the equation two real solutions.

Therefore, Statement 1, but not Statement 2, is enough to answer the question.

Report an Error

Example Question #2 : Dsq: Solving Quadratic Equations

is a positive integer.

True or false:

$x^{2} - 6x + 8 = 0$

Statement 1: is an even integer

Statement 2: x < 6

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

The quadratic expression can be factored as (x+? )(x+? ) , replacing the question marks with integers whose product is 8 and whose sum is . These integers are -2,-4 , so the equation becomes:

$x^{2} - 6x + 8 = 0$

(x-2 )(x-4 ) = 0

Set each linear binomial to 0 and solve:

$x - 2 = 0 \Rightarrow x = 2$

$x - 4 = 0 \Rightarrow x = 4$

Therefore, for the statement to be true, either x = 2 or x = 4 . Each of Statement 1 and Statement 2, taken alone, leaves other possible values of . Taken together, however, they are enough, since the only two positive even integers less than 6 are 2 and 4.

Report an Error

Example Question #161 : Algebra

is a positive integer.

True or false?

$x^{2} - 10x +24 = 0$

Statement 1: is an even integer

Statement 2: x < 8

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

The two statements together are insufficient. If both are assumed, then can be 2, 4, or 6.

If x = 4 the statement is true:

$x^{2} - 10x +24 = 0$

$4^{2} - 10 \cdot 4 +24 = 0$

16 - 40 +24 = 0

0 = 0

But if x = 2 the statement is false:

$x^{2} - 10x +24 = 0$

$2^{2} - 10 \cdot 2 +24 = 0$

4 - 20 +24 = 0

8 = 0

Report an Error

Example Question #6 : Solving Quadratic Equations

What are the solutions of $5x^{2}-4x-3$ in the most simplified form?

Possible Answers:

$\frac{2 \pm \sqrt{76}}{10}$

$\frac{1 \pm \sqrt{16}}{5}$

$\frac{1 \pm \sqrt{76}}{10}$

$\frac{2 \pm \sqrt{19}}{5}$

$\frac{1 \pm \sqrt{19}}{5}$

Correct answer:

$\frac{1 \pm \sqrt{19}}{5}$

Explanation:

This is a quadratic formula problem. Use equation $\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$ . For our problem, a=5, b=-4, c=3. Plug these values into the equation, and simplify:

$\frac{2 \pm \sqrt{(-4)^{2} - 4(5)(-3)}}{2(5)}$ $=\frac{2 \pm \sqrt{16 +60}}{10}=\frac{2 \pm \sqrt{76}}{10}=\frac{1 \pm \sqrt{19}}{5}$ . Here, we simplified the radical by $\sqrt{76}=\sqrt{4*19}=\sqrt{4}\sqrt{19}=2\sqrt{19}.$

Report an Error

All GMAT Math Resources

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

Popular Subjects

LSAT Tutors in Philadelphia, Calculus Tutors in Boston, Statistics Tutors in Houston, Algebra Tutors in Boston, Math Tutors in Phoenix, French Tutors in San Francisco-Bay Area, Statistics Tutors in Seattle, Spanish Tutors in Seattle, ACT Tutors in Houston, English Tutors in Chicago

Popular Courses & Classes

ACT Courses & Classes in New York City, SAT Courses & Classes in Seattle, Spanish Courses & Classes in Los Angeles, Spanish Courses & Classes in Chicago, GRE Courses & Classes in Houston, SAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Dallas Fort Worth, SSAT Courses & Classes in Miami, SAT Courses & Classes in Washington DC, Spanish Courses & Classes in San Diego

Popular Test Prep

MCAT Test Prep in Seattle, LSAT Test Prep in Seattle, GRE Test Prep in Seattle, GRE Test Prep in Atlanta, MCAT Test Prep in Denver, MCAT Test Prep in Chicago, ACT Test Prep in Chicago, LSAT Test Prep in Denver, MCAT Test Prep in Philadelphia, GRE Test Prep in Philadelphia

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 : Solving Quadratic Equations

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Dsq: Solving Quadratic Equations

Example Question #2 : Solving Quadratic Equations

Example Question #154 : Algebra

Example Question #2 : Dsq: Solving Quadratic Equations

Example Question #161 : Algebra

Example Question #6 : Solving Quadratic Equations

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details