Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Find roots of quadratic equation using discriminant

Study concepts, example questions & explanations for Precalculus

Create An Account

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

True or false: for a quadratic function of form ax²+ bx + c = 0, if the discriminant b²- 4ac = 0, there is exactly one real root.

Possible Answers:

False

True

Correct answer:

True

Explanation:

This is true. The discriminant b²- 4ac is the part of the quadratic formula that lives inside of a square root function. As you plug in the constants a, b, and c into b²- 4ac and evaluate, three cases can happen:

b²- 4ac > 0

b²- 4ac = 0

b²- 4ac < 0

In the first case, having a positive number under a square root function will yield a result that is a positive number answer. However, because the quadratic function includes $\pm$ , this scenario yields two real results.

In the middle case (the case of our example), $\sqrt0 = 0$ . Going back to the quadratic formula $x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$ , you can see that when everything under the square root is simply 0, then you get only $x = \frac{-b }{2a}$ , which is why you have exactly one real root.

For the final case, if b²- 4ac < 0, that means you have a negative number under a square root. This means that you will not have any real roots of the equation; however, you will have exactly two imaginary roots of the equation.

Report an Error

Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

True or false: for a quadratic function of form ax²+ bx + c = 0, if the discriminant b²- 4ac > 0, there are exactly 2 distinct real roots of the equation.

Possible Answers:

False

True

Correct answer:

True

Explanation:

b²- 4ac > 0

b²- 4ac = 0

b²- 4ac < 0

In the first case (the case of our example), having a positive number under a square root function will yield a result that is a positive number answer. However, because the quadratic function includes $\pm$ , this scenario yields two real results.

In the middle case, $\sqrt0 = 0$ . Going back to the quadratic formula $x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$ , you can see that when everything under the square root is simply 0, then you get only $x = \frac{-b }{2a}$ , which is why you have exactly one real root.

Report an Error

Example Question #3 : Find Roots Of Quadratic Equation Using Discriminant

True or false: for a quadratic function of form ax²+ bx + c = 0, if the discriminant b²- 4ac < 0, there are exactly two distinct real roots.

Possible Answers:

True

False

Correct answer:

False

Explanation:

This is false. The discriminant b²- 4ac is the part of the quadratic formula that lives inside of a square root function. As you plug in the constants a, b, and c into b²- 4ac and evaluate, three cases can happen:

b²- 4ac > 0

b²- 4ac = 0

b²- 4ac < 0

For the final case (the case of our example), if b²- 4ac < 0, that means you have a negative number under a square root. This means that you will not have any real roots of the equation; however, you will have exactly two imaginary roots of the equation.

Report an Error

Example Question #4 : Find Roots Of Quadratic Equation Using Discriminant

Use the formula b² - 4ac to find the discriminant of the following equation: 4x² + 19x - 5 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Possible Answers:

Discriminant: 0

One real root: $\frac{-19}{8}$

Discriminant: 281

Two imaginary roots: $\frac{-19\pm 281i}{8}$

Discriminant: 281

Two imaginary roots: $x = \frac{19\pm 281i}{8}$

Discriminant: 441

Two real roots: $x=\frac{1}{4}$ or x=-5

Discriminant: 441

Two real roots: $x=-\frac{1}{4}$ or x=5

Correct answer:

Discriminant: 441

Two real roots: $x=\frac{1}{4}$ or x=-5

Explanation:

In the above equation, a = 4, b = 19, and c = -5. Therefore:

b² - 4ac = (19)² - 4(4)(-5) = 361 + 80 = 441.

When the discriminant is greater than 0, there are two distinct real roots. When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zero, there are no real roots, but there are exactly two distinct imaginary roots. In this case, we have two real roots.

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

$x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$

Plugging in our values of a, b, and c, we get:

$x = \frac{-19\pm \sqrt{19^2 - 4\cdot 4\cdot -5}}{2\cdot 4}$

This simplifies to:

$x = \frac{-19\pm \sqrt{441}}{8}$

which simplifies to

$x = \frac{-19\pm21}{8}$

which gives us two answers:

$x=\frac{1}{4}$ or x=-5

These values of x are the two distinct real roots of the given equation.

Report an Error

Example Question #5 : Find Roots Of Quadratic Equation Using Discriminant

Use the formula b² - 4ac to find the discriminant of the following equation: 4x² + 12x + 10 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Possible Answers:

Discriminant: 16

Types of Roots: Two distinct real roots

Exact Roots: -1, -2

Discriminant: -16

Types of Roots: No real roots; 2 distinct imaginary roots

Exact Roots: $x=\frac{-3\pm i}{2}$

Discriminant: 304

Types of Roots: Two distinct real roots

Exact Roots: $\frac{3\pm \sqrt{19}}{2}$

Discriminant: 304

Types of Roots: Two distinct real roots

Exact Roots: $\frac{-3\pm \sqrt{19}}{2}$

Discriminant: -16

Types of Roots: No real roots; 2 distinct imaginary roots

Exact Roots: $x=\frac{3\pm i}{2}$

Correct answer:

Discriminant: -16

Types of Roots: No real roots; 2 distinct imaginary roots

Exact Roots: $x=\frac{-3\pm i}{2}$

Explanation:

In the above equation, a = 4, b = 12, and c = 10. Therefore:

b² - 4ac = (12)² - 4(4)(10) = 144 - 160 = -16.

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

$x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$

Plugging in our values of a, b, and c, we get:

$x = \frac{-12\pm \sqrt{144 - 4\cdot 4\cdot 10}}{2\cdot 4}$

This simplifies to:

$x = \frac{-12\pm \sqrt{-16}}{8}$

$x=\frac{-12\pm 4i}{8}$

$x=\frac{-3\pm i}{2}$

In other words, our two distinct imaginary roots are $\frac{-3+i}{2}$ and $\frac{-3-i}{2}$

Report an Error

Example Question #6 : Find Roots Of Quadratic Equation Using Discriminant

Use the formula b² - 4ac to find the discriminant of the following equation: -3x² + 6x - 3 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Possible Answers:

Discriminant: 72

Two distinct real roots: $x=1\pm \sqrt{2}$

Discriminant: -72

Two distinct imaginary roots: $x=1\pm 2i$

Discriminant: 72

Two distinct real roots: $x=-1\pm \sqrt{2}$

Discriminant: 0

One real root: x = 1

Discriminant: 0

One real root: x = -1

Correct answer:

Discriminant: 0

One real root: x = 1

Explanation:

In the above equation, a = -3, b = 6, and c = -3. Therefore:

b² - 4ac = (6)² - 4(-3)(-3) = 36 - 36 = 0.

Finally, we use the quadratic function to find these exact root. The quadratic formula is:

$x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$

Plugging in our values of a, b, and c, we get:

$x = \frac{-6\pm \sqrt{36 - 4\cdot -3\cdot -3}}{2\cdot -3}$

This simplifies to:

$x = \frac{-6\pm \sqrt{36 - 36}}{-6}$

which simplifies to

$x= \frac{-6\pm0}{-6}$

which gives us one answer: x = 1

This value of x is the one distinct real root of the given equation.

Report an Error

Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

Use the formula b² - 4ac to find the discriminant of the following equation: x² + 5x + 4 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Possible Answers:

Discriminant: 41

Two imaginary roots: $x=\frac{-5\pm 41i}{2}$

Discriminant: 0

One real root: $x=\frac{-5}{2}$

Discriminant: 9

Two real roots: x = -1 or x = -4

Discriminant: 41

Two imaginary roots: $x=\frac{5\pm 41i}{2}$

Discriminant: 9

Two real roots: x = 1 or x = 4

Correct answer:

Discriminant: 9

Two real roots: x = -1 or x = -4

Explanation:

In the above equation, a = 1, b = 5, and c = 4. Therefore:

b² - 4ac = (5)² - 4(1)(4) = 25 - 16 = 9.

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

$x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$

Plugging in our values of a, b, and c, we get:

$x = \frac{-5\pm \sqrt{5^2 - 4\cdot 1\cdot 4}}{2\cdot 1}$

This simplifies to:

$x = \frac{-5\pm \sqrt{9}}{2}$

which simplifies to

$x=\frac{-5\pm3}{2}$

which gives us two answers:

x = -1 or x = -4

These values of x are the two distinct real roots of the given equation.

Report an Error

Example Question #8 : Find Roots Of Quadratic Equation Using Discriminant

Use the formula b² - 4ac to find the discriminant of the following equation: -x² + 3x - 3 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Possible Answers:

Discriminant: -8

Two imaginary roots: $x=\frac{-3\pm\sqrt{3}i}{2}$

Discriminant: 0

One real root: $x=\frac{3}{2}$

Discriminant: -21

Two imaginary roots: $x=\frac{3\pm\sqrt{21}i}{2}$

Discriminant: -8

Two imaginary roots: $x=\frac{3\pm\sqrt{3}i}{2}$ .

Discriminant: -21

Two imaginary roots: $x=\frac{-3\pm\sqrt{21}i}{2}$

Correct answer:

Discriminant: -8

Two imaginary roots: $x=\frac{3\pm\sqrt{3}i}{2}$ .

Explanation:

In the above equation, a = -1, b = 3, and c = -3. Therefore:

b² - 4ac = (3)² - 4(-1)(-3) = 9 - 12 = -3.

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

$x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$

Plugging in our values of a, b, and c, we get:

$x = \frac{-3\pm \sqrt{3^2 - 4\cdot -1\cdot -3}}{2\cdot -1}$

This simplifies to:

$x = \frac{-3\pm \sqrt{-3}}{-2}$

Because $\sqrt{-3}=\sqrt{(-1)(3)}=\sqrt{3}i$ , this simplifies to $x=\frac{3\pm\sqrt{3}i}{2}$ . In other words, our two distinct imaginary roots are $x=\frac{3+\sqrt{3}i}{2}$ and $x=\frac{3-\sqrt{3}i}{2}$

Report an Error

Example Question #9 : Find Roots Of Quadratic Equation Using Discriminant

Use the formula b² - 4ac to find the discriminant of the following equation: x² + 2x + 10 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Possible Answers:

Discriminant: 36

Two real roots: x = 5 or x = -7

Discriminant: 36

Two real roots: x = -5 or x = 7

Discriminant: -36

Two imaginary roots: $1\pm3i$

Discriminant: 0

One real root: x = -1

Discriminant: -36

Two imaginary roots: $-1\pm3i$

Correct answer:

Discriminant: -36

Two imaginary roots: $-1\pm3i$

Explanation:

In the above equation, a = 1, b = 2, and c = 10. Therefore:

b² - 4ac = (2)² - 4(1)(10) = 4 - 40 = -36.

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

$x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$

Plugging in our values of a, b, and c, we get:

$x = \frac{-2\pm \sqrt{2^2 - 4\cdot 1\cdot 10}}{2\cdot 1}$

This simplifies to:

$x = \frac{-2\pm \sqrt{-36}}{2}$

Because $\sqrt{-36}=\sqrt{(-1)(36)}=6i$ , this simplifies to $x=\frac{-2\pm6i}{2}$ . We can further simplify this to $-1\pm3i$ . In other words, our two distinct imaginary roots are -1+3i and -1-3i .

Report an Error

Example Question #10 : Find Roots Of Quadratic Equation Using Discriminant

Use the formula b² - 4ac to find the discriminant of the following equation: x² + 8x + 16 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

Possible Answers:

Discriminant: 0

One real root: x = 0

Discriminant: 0

One real root: x = 4

Discriminant: 128

Two distinct real roots: $x=-4\pm4\sqrt{2}$

Discriminant: 0

One real root: x = -4

Discriminant: 72

Two distinct real roots: $x=4\pm4\sqrt2$

Correct answer:

Discriminant: 0

One real root: x = -4

Explanation:

In the above equation, a = 1, b = 8, and c = 16. Therefore:

b² - 4ac = (8)² - 4(1)(16) = 64 - 64 = 0.

Finally, we use the quadratic function to find these exact root. The quadratic formula is:

$x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$

Plugging in our values of a, b, and c, we get:

$x = \frac{-8\pm \sqrt{8^2 - 4\cdot 1\cdot 16}}{2\cdot 1}$

This simplifies to:

$x = \frac{-8\pm \sqrt{0}}{2}$

which simplifies to

x=-4

This value of x is the one distinct real root of the given equation.

Report an Error

View Pre-Calculus Tutors

A K M Shabab
Certified Tutor

Carleton University, Bachelor of Engineering, Computer Engineering, General.

View Pre-Calculus Tutors

Andrew
Certified Tutor

University of Wisconsin-Stevens Point, Bachelor of Science, Mathematics. University of Wisconsin-Oshkosh, Master of Science, ...

View Pre-Calculus Tutors

Yuliya
Certified Tutor

University of Maryland-College Park, Bachelor of Science, Mathematics. George Washington University, Masters in Education, Ed...

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Statistics Tutors in Houston, Physics Tutors in Washington DC, SSAT Tutors in Los Angeles, ISEE Tutors in San Diego, Biology Tutors in Los Angeles, ISEE Tutors in Los Angeles, Spanish Tutors in Washington DC, Chemistry Tutors in San Diego, GMAT Tutors in Houston, Physics Tutors in Miami

Popular Courses & Classes

LSAT Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Atlanta, ACT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Houston, GRE Courses & Classes in Atlanta, MCAT Courses & Classes in Houston, ACT Courses & Classes in New York City, ACT Courses & Classes in San Diego, Spanish Courses & Classes in Boston, Spanish Courses & Classes in Phoenix

Popular Test Prep

SAT Test Prep in Phoenix, ACT Test Prep in Houston, GRE Test Prep in Phoenix, GMAT Test Prep in San Francisco-Bay Area, ISEE Test Prep in Atlanta, ACT Test Prep in San Francisco-Bay Area, ACT Test Prep in Chicago, ACT Test Prep in New York City, SAT Test Prep in Washington DC, GMAT Test Prep in Denver

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)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Precalculus : Find roots of quadratic equation using discriminant

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #3 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #4 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #5 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #6 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #8 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #9 : Find Roots Of Quadratic Equation Using Discriminant

Example Question #10 : Find Roots Of Quadratic Equation Using Discriminant

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: