Call Now to Set Up Tutoring:

(888) 888-0446

Algebra II : Solving Quadratic Equations

Study concepts, example questions & explanations for Algebra II

Create An Account

All Algebra II Resources

10 Diagnostic Tests 630 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #11 : Completing The Square

Solve the following equation by completing the square.

x^2 - 4x - 16 = 5

Possible Answers:

$x = 2 \pm\sqrt{21}$

$x = \left \{ -3, 7\right \}$

x = 7

$x = \left \{ -7, 3\right \}$

$x = 2 \pm\sqrt{29}$

Correct answer:

$x = \left \{ -3, 7\right \}$

Explanation:

Starting with the original equation, we move the constant term of the quadratic over to the other side, so we can set up our "completing the square."

x^2 - 4x - 16 = 5

x^2 - 4x - 16 + 16 = 5 + 16

x^2 - 4x = 21

Since the x^2 coefficient is already 1, we don't have to do any division. We can go straight to completing the square by dividing the x coefficient by 2, squaring the result, and adding that result to both sides.

$\frac{-4}{2} = -2$

(-2)^2 = 4

x^2 - 4x + 4 = 21 + 4

x^2 - 4x + 4 = 25

Since the left side is now a perfect square, we can rewrite it as a squared binomial.

(x-2)^2 = 25

Now take the square root of both sides and solve for x.

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

$x-2 = \pm 5$

$x = 2 \pm5$

Report an Error

Example Question #12 : Completing The Square

Rewrite the follow parabola equation in vertex form:

y = 4x^2 + 24x + 16

Possible Answers:

y = 4(x+3)^2 - 36

y = 4(x+3)^2 - 20

y = 4(x+3)^2

y = (x+3)^2 - 5

y = 4(x+3)^2 - 5

Correct answer:

y = 4(x+3)^2 - 20

Explanation:

We start by moving the constant term of the quadratic over to the other side of the equation, in order to set up our "completing the square" form.

y = 4x^2 + 24x + 16

y - 16 = 4x^2 + 24x + 16 -16

y -16 = 4x^2 + 24x

Next we divide both sides by 4 so that the x^2 coefficient will be 1. That will allow us to complete the square.

$\frac{y-16}{4} = \frac{4x^2 + 24x}{4}$

$\frac{y}{4} - 4 = x^2 + 6 x$

Now we are ready to complete the square. We divide the x coefficient by 2, square the result, and add that result to both sides.

$\frac{6}{2} = 3$

3^2 = 9

$\frac{y}{4} - 4 + 9 = x^2 + 6 x + 9$

$\frac{y}{4} +5 = x^2 + 6 x + 9$

Because the right side is now a perfect square, we can rewrite it as a squared binomial.

$\frac{y}{4} + 5 = (x+3)^2$

To finish, all we have to do now is solve for y. We'll subtract 5 to both sides, then multiply by 4.

$\frac{y}{4} + 5 -5 = (x+3)^2 - 5$

$\frac{y}{4} = (x+3)^2 - 5$

$4*\frac{y}{4} = ((x+3)^2 - 5)*4$

y = 4(x+3)^2 - 5*4

y = 4(x+3)^2 - 20

Report an Error

Example Question #13 : Completing The Square

Solve by completing the square.

$x^{2}-8x+21=6$

Possible Answers:

x=1,3

x=5

x=-5,5

x=3,5

x=1,5

Correct answer:

x=3,5

Explanation:

The first step is to make sure the x² term has a coefficient of 1. Since we have that, we move onto the next step.

Next, move the "loose" number over to the other side.

$x^{2}-8x=-15$

Now divide the coefficient of the x term by 2 (don't forget the sign!). Add the square of this number to both sides.

$-8x \rightarrow -\frac{8}{2}=-4 \rightarrow (-4)^{2}=16$

$x^{2}-8x+16=-15+16$

Simplify:

$x^{2}-8x+16=1$

Now the left side of the equation can be simplified to a squared factor. The factor that will be squared is going to be x plus the original x coefficient divided by 2 as we calculated above.

$(x-4)^{2}=1$

Take the square root of both sides:

$x-4=\pm 1$

Solve for x:

$x-4=1 \rightarrow x=5$

$x-4=-1 \rightarrow x=3$

x=3,5

Report an Error

Example Question #81 : Solving Quadratic Equations

2x^2-5x-8=0

Using the above equation, what should the next step look like when completing the square?

Possible Answers:

2x^2=5x+8

2x^2-5x=8

x^2-(5/2)x-8=0

2x^2-5x=-8

Correct answer:

2x^2-5x=8

Explanation:

The first step when completing the square is to move the constant to the other side of the equation by subtracting from both sides. Don't forget the sign!

Report an Error

Example Question #82 : Solving Quadratic Equations

2x^2-5x=8

In the above equation, what must next be done to both sides of the equation when completing the square?

Possible Answers:

Divide by 2.

Divide by -5.

Divide by 5.

Add 5x.

Subtract 5x.

Correct answer:

Divide by 2.

Explanation:

When completing the square, the lead coefficient should be one. To achieve this, divide both sides of the equation by the coefficient of the squared term.

$(2x^2-5x)\div2=x^2-\frac{5}{2}x$

and

$8\div2=4$

This leaves you with the equation $x^2-\frac{5}{2}x=4$

Report an Error

Example Question #16 : Completing The Square

$x^2-\frac{5}{2}x=4$

In the above equation, what should be added to both sides of the equation in order to complete the square?

Possible Answers:

$-\frac{5}{4}$

$\frac{5}{2}$

$\frac{5}{4}$

$-\frac{25}{16}$

$\frac{25}{16}$

Correct answer:

$\frac{25}{16}$

Explanation:

a. Once the variables are on the left, the constant is on the right, and the lead coefficient is 1, you will create a perfect square trinomial on the left side of the equation. Do this by starting with the coefficient of the x term.

$\frac{5}{2}$

b. Divide this by 2.

$\frac{5}{2}\div2=\frac{5}{4}$

c. Square this term.

$(\frac{5}{4})^2=\frac{25}{16}$

d. Add the result to both sides of the equation.

$x^2-\frac{5}{2}x+\frac{25}{16}=\frac{89}{16}$

e. The expression on the left side of the equation is now a perfect square trinomial and can be factored to:

$(x-\frac{5}{4})^2=\frac{89}{16}$

Report an Error

Example Question #87 : Solving Quadratic Equations

$(x-\frac{5}{4})^2=\frac{89}{16}$

Finish solving the above equation by completing the square.

Possible Answers:

$\frac{47}{2}$

$x=\frac{5}{4}\pm\frac{\sqrt89}{4}$

$x=\frac{5}{4}+\frac{\sqrt89}{4}$

$x=\frac{5}{4}-\frac{\sqrt89}{4}$

Correct answer:

$x=\frac{5}{4}\pm\frac{\sqrt89}{4}$

Explanation:

a. Take the square root of both sides. Don't forget to account for both positive and negative answers.

$\sqrt{(x-\frac{5}{4})^2}=\pm\sqrt{\frac{89}{16}}$

b. Simplify both sides

$x-\frac{5}{4}=\pm\frac{\sqrt{89}}{4}$

c. Isolate the variable by adding 5/4 to both sides.

$x=\frac{5}{4}\pm\frac{\sqrt{89}}{4}$

Report an Error

Example Question #83 : Solving Quadratic Equations

3x^2+12x=-5

Using the above equation, what should the next step look like when completing the square?

Possible Answers:

$x^2+4x=-\frac{5}{3}$

x^2+4x=-5

x^2+12x=-5

3x^2=-12x-5

Correct answer:

$x^2+4x=-\frac{5}{3}$

Explanation:

a. Once the variables are on the left side of the equation and the constant is on the right, make the lead coefficient 1 by dividing both sides of the equation by the coefficient of the squared term.

$(3x^2+12x)\div3=-5\div3$

b. Simplify

$x^2+4x=-\frac{5}{3}$

Report an Error

Example Question #89 : Solving Quadratic Equations

$x^2+4x=-\frac{5}{3}$

In the above equation, what must next be done to both sides of the equation when completing the square?

Possible Answers:

Add 4.

Divide by 4.

Subtract 4.

Add 4x.

Correct answer:

Add 4.

Explanation:

To complete the square, you must create a perfect square trinomial on the left side of the equation.

Do this by starting with the coefficient of the x term.

b. Divide this by 2.

$4\div2=2$

c. Square this term.

2^2=4

d. Add the result to both sides of the equation.

$x^2+4x+4=4-\frac{5}{3}$

e. The expression on the left side of the equation is now a perfect square trinomial and can be factored to:

$(x+2)^2=\frac{7}{3}$

Report an Error

Example Question #90 : Solving Quadratic Equations

$(x+2)^2=\frac{7}{3}$

Finish solving the above equation by completing the square.

Possible Answers:

$x=2+\sqrt{\frac{7}{3}}$

$x=2\pm\sqrt{\frac{7}{3}}$

$x=-2\pm\sqrt{\frac{7}{3}}$

$x=-2-\sqrt{\frac{7}{3}}$

Correct answer:

$x=-2\pm\sqrt{\frac{7}{3}}$

Explanation:

a. Take the square root of both sides. Don't forget to account for both positive and negative answers.

$\sqrt{(x+2)^2}=\pm\sqrt{\frac{7}{3}}$

b. Simplify both sides, if possible.

$x+2=\pm\sqrt{\frac{7}{3}}$

c. Isolate the variable by subtracting 2 from both sides.

$x=-2\pm\sqrt{\frac{7}{3}}$

Report an Error

View Algebra 2 Tutors

Robert Edward
Certified Tutor

University of Calgary, Master of Engineering, Chemistry. University of Ottawa, Bachelor of Engineering, Civil Engineering.

View Algebra 2 Tutors

Shakeel
Certified Tutor

View Algebra 2 Tutors

Sharon
Certified Tutor

Tabor College, Bachelor of Science, Mathematics Teacher Education. Tabor College, Masters in Education, Education.

All Algebra II Resources

10 Diagnostic Tests 630 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Physics Tutors in San Diego, Statistics Tutors in Chicago, French Tutors in Phoenix, Statistics Tutors in Miami, Spanish Tutors in Chicago, Algebra Tutors in Los Angeles, Math Tutors in New York City, GRE Tutors in Dallas Fort Worth, Biology Tutors in Dallas Fort Worth, English Tutors in San Diego

Popular Courses & Classes

LSAT Courses & Classes in Philadelphia, SAT Courses & Classes in Houston, GMAT Courses & Classes in San Diego, MCAT Courses & Classes in Houston, ISEE Courses & Classes in Atlanta, MCAT Courses & Classes in Boston, SAT Courses & Classes in New York City, Spanish Courses & Classes in Boston, ISEE Courses & Classes in San Diego, LSAT Courses & Classes in Dallas Fort Worth

Popular Test Prep

GRE Test Prep in Boston, SAT Test Prep in Phoenix, LSAT Test Prep in Philadelphia, SSAT Test Prep in Miami, MCAT Test Prep in New York City, SAT Test Prep in San Diego, ISEE Test Prep in Miami, GRE Test Prep in Seattle, GMAT Test Prep in Denver, GRE Test Prep in San Diego

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

Algebra II : Solving Quadratic Equations

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #11 : Completing The Square

Example Question #12 : Completing The Square

Example Question #13 : Completing The Square

Example Question #81 : Solving Quadratic Equations

Example Question #82 : Solving Quadratic Equations

Example Question #16 : Completing The Square

Example Question #87 : Solving Quadratic Equations

Example Question #83 : Solving Quadratic Equations

Example Question #89 : Solving Quadratic Equations

Example Question #90 : Solving Quadratic Equations

All Algebra II Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: