# Connexions

You are here: Home » Content » Quadratic Concepts -- The Quadratic Formula

### Lenses

What is a lens?

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

#### Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• Bookshare

This module is included inLens: Bookshare's Lens
By: Bookshare - A Benetech InitiativeAs a part of collection: "Advanced Algebra II: Conceptual Explanations"

"DAISY and BRF versions of this collection are available."

Click the "Bookshare" link to see all content affiliated with them.

• Featured Content

This module is included inLens: Connexions Featured Content
By: ConnexionsAs a part of collection: "Advanced Algebra II: Conceptual Explanations"

"This is the "concepts" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing about […]"

Click the "Featured Content" link to see all content affiliated with them.

Click the tag icon to display tags associated with this content.

#### Also in these lenses

• Busbee's Math Materials

This module is included inLens: Busbee's Math Materials Lens
By: Kenneth Leroy BusbeeAs a part of collection: "Advanced Algebra II: Conceptual Explanations"

Click the "Busbee's Math Materials" link to see all content selected in this lens.

Click the tag icon to display tags associated with this content.

### Recently Viewed

This feature requires Javascript to be enabled.

### Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Module by: Kenny M. Felder. E-mail the author

Summary: This module discusses the quadratic formula.

In "Solving Quadratic Equations by Completing the Square" I talked about the common mathematical trick of solving a problem once, using letters instead of numbers, and then solving specific problems by plugging numbers into a general solution.

In the text, you go through this process for quadratic equations in general. The definition of a quadratic equation is any equation that can be written in the form:

ax 2 + bx + c = 0 ax 2 + bx + c = 0 size 12{ ital "ax" rSup { size 8{2} } + ital "bx"+c=0} {}
(1)

where a0a0 size 12{a <> 0} {}. By completing the square on this generic equation, you arrive at the quadratic formula:

x = b ± b 2 4 ac 2a x = b ± b 2 4 ac 2a size 12{x= { { - b +- sqrt {b rSup { size 8{2} } - 4 ital "ac"} } over {2a} } } {}
(2)

This formula can then be used to solve any quadratic equation, without having to complete the square each time. To see how this formula works, let us return to the previous problem:

9x 2 54 x + 80 = 0 9x 2 54 x + 80 = 0 size 12{9x rSup { size 8{2} } - "54"x+"80"=0} {}
(3)

In this case, a=9a=9 size 12{a=9} {}, b=54b=54 size 12{b= - "54"} {}, and c=80c=80 size 12{c="80"} {}. So the quadratic formula tells us that the answers are:

x = ( 54 ) ± ( 54 ) 2 4 ( 9 ) ( 80 ) 2 ( 9 ) x = ( 54 ) ± ( 54 ) 2 4 ( 9 ) ( 80 ) 2 ( 9 ) size 12{x= { { - $$- "54"$$ +- sqrt { $$- "54"$$ rSup { size 8{2} } - 4 $$9$$ $$"80"$$ } } over {2 $$9$$ } } } {}
(4)

We’ll use a calculator here rather than squaring 54 by hand....

x = 54 ± 2916 2880 18 = 54 ± 36 18 = 54 ± 6 18 = 9 ± 1 3 x = 54 ± 2916 2880 18 = 54 ± 36 18 = 54 ± 6 18 = 9 ± 1 3 size 12{x= { {"54" +- sqrt {"2916" - "2880"} } over {"18"} } = { {"54" +- sqrt {"36"} } over {"18"} } = { {"54" +- 6} over {"18"} } = { {9 +- 1} over {3} } } {}
(5)

So we find that the two answers are 103103 size 12{ { {"10"} over {3} } } {} and 8383 size 12{ { {8} over {3} } } {}, which are the same answers we got by completing the square.

Using the quadratic formula is usually faster than completing the square, though still slower than factoring. So, in general, try to factor first: if you cannot factor, use the quadratic formula.

So why do we learn completing the square? Two reasons. First, completing the square is how you derive the quadratic formula. Second, completing the square is vital to graphing quadratic functions, as you will see a little further on in the chapter.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

### Reuse / Edit:

Reuse or edit module (?)

#### Check out and edit

If you have permission to edit this content, using the "Reuse / Edit" action will allow you to check the content out into your Personal Workspace or a shared Workgroup and then make your edits.

#### Derive a copy

If you don't have permission to edit the content, you can still use "Reuse / Edit" to adapt the content by creating a derived copy of it and then editing and publishing the copy.