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 : Solving Quadratic Equations

Solve for .

x^2+4x+9=15

Possible Answers:

$x=-4\pm2\sqrt{10}$

$x=\pm 3$

$x=4\pm4\sqrt{10}$

$x=-2\pm \sqrt{10}$

Correct answer:

$x=-2\pm \sqrt{10}$

Explanation:

Solve using the quadratic formula:

$ax^2+bx+c=0;\ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

x^2+4x+9=15

$x=\frac{-4\pm\sqrt{4^2-(4)(1)(-6)}}{(2)(1)}$

$x=\frac{-4\pm\sqrt{16+24}}{2}$

$x=\frac{-4\pm\sqrt{40}}{2}$

$x=\frac{-4\pm2\sqrt10}{2}$

$x=-2\pm\sqrt{10}$

Report an Error

Example Question #2 : Solving Quadratic Equations

Multiply: $(x^{2} - 5x + 3) (x - 2)$

Possible Answers:

$x^{3} - 7 x^{2}+7x - 6$

$x^{3} +3 x^{2}+13x - 6$

$x^{3} +3 x^{2}+7x - 6$

$x^{3} - 7 x^{2}+13x - 6$

$x^{3} +7 x^{2}+10x - 6$

Correct answer:

$x^{3} - 7 x^{2}+13x - 6$

Explanation:

Distribute:

$(x^{2} - 5x + 3) (x - 2)$

$=x^{2}(x - 2) - 5x(x - 2) + 3(x - 2)$

$=x^{2}\cdot x - x^{2}\cdot 2- 5x \cdot x +5x \cdot 2 + 3 \cdot x - 3 \cdot 2$

$=x^{3} - 2 x^{2}- 5 x^{2} +10x + 3 x - 6$

$=x^{3} - 7 x^{2}+13x - 6$

Report an Error

Example Question #3 : Solving Quadratic Equations

Consider the equation $x^{2} -Bx + B = 0$ . For what value(s) of does the equation have two real solutions?

Possible Answers:

B< -4

B< -4 or B >4

B< 0 or B >4

B< -4 or B >0

B >4

Correct answer:

B< 0 or B >4

Explanation:

The discriminant of the expression $x^{2} -Bx + B$ is $\left (-B \right )^{2} - 4 \cdot1\cdot B = B^{2} -4B$ . For the equation $x^{2} -Bx + B = 0$ to have two real solutions, this discriminant must be positive, so:

$B^{2} -4B > 0$

$B \left (B -4 \right ) > 0$

One of two things happens:

Case 1:

B> 0 and B -4 >0

B> 0 and B >4

But this is the same as simply saying B >4

Case 2:

B< 0 and B -4 <0

B< 0 and B <4

But this is the same as simply saying B< 0

Therefore, the equation has two real solutions if and only if either B< 0 or B >4

Report an Error

Example Question #4 : Solving Quadratic Equations

Find all real solutions for :

$x^{4} + 9x ^{2}- 10 = 0$

Possible Answers:

$x \in \left \{-\sqrt{10},-1,1, \sqrt{10} \right \}$

$x \in \left \{1, \sqrt{10} \right \}$

$x \in \left \{ -1,1 \right \}$

$x \in \left \{-\sqrt{10}, \sqrt{10} \right \}$

The equation has no real solution.

Correct answer:

$x \in \left \{ -1,1 \right \}$

Explanation:

Substitute $u = x^{2}$ , and, subsequently, $u^{2} =\left ( x^{2} \right )^{2} = x^{4}$ , and solve the resulting quadratic equation.

$x^{4} + 9x ^{2}- 10 = 0$

$u^{2} + 9u- 10 = 0$

We can rewrite the quadratic expression as (x + ?) (x + ?) , where the question marks are replaced with integers whose product is and whose sum is 9; the integers are -1,10 :

(u-1)(u+10)= 0

Set each factor to zero and solve for ; then substitute back and solve for :

u - 1 = 0

u = 1

$x^{2}= 1$

$x = -1 \textrm{ or } x = 1$

u + 10 = 0

u = -10

$x^{2}= -10$ , which has no real solution.

Therefore, the solution set is $\left \{ -1,1\right \}$

Report an Error

Example Question #1 : Solving Quadratic Equations

What is the minimum value of the function $f(x) = x^{2} - 7x + 7$ for all real values of ?

Possible Answers:

$-19 \frac{1}{4}$

$-3\frac{1}{2}$

$-5 \frac{1}{4}$

f(x) does not have a minimum value.

Correct answer:

$-5 \frac{1}{4}$

Explanation:

We find the -coordinate of the vertex of the parabola for f(x) . First, we find its -coordinate using the formula

$x = -\frac{b}{2a}$ , setting a = 1, b = -7 .

$x = -\frac{b}{2a} = -\frac{-7}{2 \cdot 1} = \frac{7}{2}$

$f \left ( \frac{7}{2} \right)$ is the -coordindate of the vertex, and, subsequently, the minimum value of f(x) :

$f(x) = x^{2} - 7x + 7$

$f\left (\frac{7}{2} \right ) = \left (\frac{7}{2} \right ) ^{2} - 7\cdot \left (\frac{7}{2} \right )+ 7$

$f\left (\frac{7}{2} \right ) = \frac{49}{4} - \frac{49}{2} + 7$

$f\left (\frac{7}{2} \right ) = \frac{49}{4} - \frac{98}{4} + \frac{28}{4}$

$f\left (\frac{7}{2} \right ) = -\frac{21}{4} = -5 \frac{1}{4}$

Report an Error

Example Question #2 : Solving Quadratic Equations

Define an operation as follows:

For all real numbers a,b ,

$a \; \bigstar \; b = a^{2} - ab$

Solve for : $x\; \bigstar\; 4 = 21$

Possible Answers:

x = -3

$x = -3 \textrm{ or }x = 7$

$x = -1\frac{1}{4}$

x = 7

$x = -1\frac{1}{4} \textrm{ or }x = 1\frac{1}{4}$

Correct answer:

$x = -3 \textrm{ or }x = 7$

Explanation:

Subsitute a = x, b = 4 in the defintion, and set it equal to 21:

$a \; \bigstar \; b = a^{2} - ab$

$x \; \bigstar \; 4 = 21$

$x^{2} - x \cdot 4 = 21$

$x^{2} - 4x = 21$

This sets up a quadratic equation, Move all terms to the left, factor the expression, set each factor to 0, and solve separately.

$x^{2} - 4x - 21 = 0$

(x-7)(x+3) = 0

$x-7 = 0 \Rightarrow x = 7$

$x+3 = 0 \Rightarrow x = -3$

The solution set is $\left \{ -3, 7\right \}$

Report an Error

Example Question #4 : Solving Quadratic Equations

Solve for ;

(x+9)(x+4)= 28x

Possible Answers:

$x=-18\textrm{ or } x=-2$

$x=-9\textrm{ or } x=-4$

$x=2\textrm{ or } x=18$

$x=-12\textrm{ or } x=-3$

$x=3 \textrm{ or } x=12$

Correct answer:

$x=3 \textrm{ or } x=12$

Explanation:

First, rewrite the quadratic equation in standard form by FOILing out the product on the left, then colleciting all of the terms on the left side:

(x+9)(x+4)= 28x

$x^{2} + 4x + 9x + 9\cdot 4= 28x$

$x^{2} + 13x + 36= 28x$

$x^{2} + 13x + 36- 28x = 28x - 28 x$

$x^{2} -15x + 36 =0$

Now factor the quadratic expression $x^{2} -15x + 36$ to two binomial factors (x + ?)(x + ?) , replacing the question marks with two integers whose product is 36 and whose sum is . These numbers are -12,-3 , so:

(x -12)(x -3) = 0

$x-12 =0 \Rightarrow x= 12$

$x-3 =0 \Rightarrow x= 3$

The solution set is $\left \{ 3,12 \right \}$

Report an Error

Example Question #1 : Solving Quadratic Equations

Solve for :

$x ^{2} + 16 = 10x$

Possible Answers:

x= 4

$x= 6 \textrm{ or }x=10$

$x= -6 \textrm{ or }x=-10$

$x= -2 \textrm{ or }x=-8$

$x= 2 \textrm{ or }x=8$

Correct answer:

$x= 2 \textrm{ or }x=8$

Explanation:

First, rewrite the quadratic equation in standard form by moving all nonzero terms to the left:

$x ^{2} + 16 = 10x$

$x ^{2} + 16- 10x = 10x - 10x$

$x ^{2} - 10x+ 16 = 0$

Now factor the quadratic expression $x ^{2}- 10x + 16$ into two binomial factors (x + ?)(x + ?) , replacing the question marks with two integers whose product is 16 and whose sum is . These numbers are -8,-2 , so:

$x ^{2}- 10x + 16 = 0$

(x -2)(x -8) = 0

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

$x-8 = 0 \Rightarrow x=8$

The solution set is $\left \{ 2,8\right \}$ .

Report an Error

Example Question #2 : Solving Quadratic Equations

Find the roots of x^2+x-2=0. Separate the answers with a comma.

Possible Answers:

2,1

-2,-1

2,-1

-2,1

Correct answer:

-2,1

Explanation:

This can be found by factoring the equation. Doing this we get

x^2+x-2=(x+2)(x-1)=0

We can solve this equation happen when x+2=0 or x-1=0. So the roots are -2,1 .

Report an Error

Example Question #395 : Algebra

How many real solutions and how many imaginary solutions are there to the following quadratic equation?

$2x ^{2} + 4x = 17$

Possible Answers:

No real solutions and one imaginary solution.

Two real solutions and no imaginary solutions.

One real solution and no imaginary solutions.

No real solutions and two imaginary solutions.

One real solution and one imaginary solution.

Correct answer:

Two real solutions and no imaginary solutions.

Explanation:

Write the equation in standard form:

$2x ^{2} + 4x = 17$

$2x ^{2} + 4x - 17= 17 - 17$

$2x ^{2} + 4x - 17=0$

Evaluate the discriminant $b^{2} - 4ac$ , using a = 2, b = 4, c = -17 .

$b^{2} - 4ac = 4 ^{2} - 4 \cdot 2 \cdot (-17 ) = 16- (-136 ) = 16 + 136 = 152 > 0$

The discriminant is positive, so the equation has two distinct real solutions.

Report an Error

All GMAT Math Resources

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

Popular Subjects

Biology Tutors in Seattle, GRE Tutors in Atlanta, French Tutors in Miami, Math Tutors in Boston, ISEE Tutors in Miami, Reading Tutors in San Diego, LSAT Tutors in Dallas Fort Worth, GRE Tutors in Dallas Fort Worth, SSAT Tutors in Chicago, Reading Tutors in Phoenix

Popular Courses & Classes

SAT Courses & Classes in Philadelphia, LSAT Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in Phoenix, GRE Courses & Classes in Chicago, ACT Courses & Classes in Washington DC, GMAT Courses & Classes in San Francisco-Bay Area, GMAT Courses & Classes in Boston, ACT Courses & Classes in Phoenix, ISEE Courses & Classes in San Diego, LSAT Courses & Classes in Denver

Popular Test Prep

GRE Test Prep in Denver, ACT Test Prep in Washington DC, ISEE Test Prep in Atlanta, ISEE Test Prep in New York City, ISEE Test Prep in Boston, SSAT Test Prep in Denver, LSAT Test Prep in San Francisco-Bay Area, GRE Test Prep in Washington DC, SSAT Test Prep in Seattle, SAT Test Prep in Houston

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 : Solving Quadratic Equations

Example Question #2 : Solving Quadratic Equations

Example Question #3 : Solving Quadratic Equations

Example Question #4 : Solving Quadratic Equations

Example Question #1 : Solving Quadratic Equations

Example Question #2 : Solving Quadratic Equations

Example Question #4 : Solving Quadratic Equations

Example Question #1 : Solving Quadratic Equations

Example Question #2 : Solving Quadratic Equations

Example Question #395 : Algebra

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details