Connexions

You are here: Home » Content » Advanced Algebra II: Conceptual Explanations » Different Types of Solutions to Quadratic Equations

• How to Use Advanced Algebra II

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 collection is included inLens: Bookshare's Lens
By: Bookshare - A Benetech Initiative

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

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

• Featured Content

This collection is included inLens: Connexions Featured Content
By: Connexions

"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 collection is included inLens: Busbee's Math Materials Lens
By: Kenneth Leroy Busbee

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.

Inside Collection (Textbook):

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

Different Types of Solutions to Quadratic Equations

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

Summary: This module discusses the different types of solutions to quadratic equations.

The heart of the quadratic formula is the part under the square root: b24acb24ac size 12{b rSup { size 8{2} } - 4 ital "ac"} {}. This part is so important that it is given its own name: the discriminant. It is called this because it discriminates the different types of solutions that a quadratic equation can have.

Common Error

Students often think that the discriminant is b24acb24ac size 12{ sqrt {b rSup { size 8{2} } - 4 ital "ac"} } {}. But the discriminant is not the square root, it is the part that is under the square root:

Discriminant = b 2 4 ac Discriminant = b 2 4 ac size 12{"Discriminant"=b rSup { size 8{2} } - 4 ital "ac"} {}
(1)
It can often be computed quickly and easily without a calculator.

Why is this quantity so important? Consider the above example, where the discriminant was 36. This means that we wound up with ±6±6 size 12{ +- 6} {} in the numerator. So the problem had two different, rational answers: 223223 size 12{2 { { size 8{2} } over { size 8{3} } } } {} and 313313 size 12{3 { { size 8{1} } over { size 8{3} } } } {}.

Now, consider x2+3x+1=0x2+3x+1=0 size 12{x rSup { size 8{2} } +3x+1=0} {}. In this case, the discriminant is 324(1)(1)=5324(1)(1)=5 size 12{3 rSup { size 8{2} } - 4 $$1$$ $$1$$ =5} {}. We will end up with ±5±5 size 12{ +- sqrt {5} } {} in the numerator. There will still be two answers, but they will be irrational—they will be impossible to express as a fraction without a square root.

4x220x+25=04x220x+25=0 size 12{4x rSup { size 8{2} } - "20"x+"25"=0} {}. Now the discriminant is 2024(4)(25)=400400=02024(4)(25)=400400=0 size 12{"20" rSup { size 8{2} } - 4 $$4$$ $$"25"$$ ="400" - "400"=0} {}. We will end up with ±0 in the numerator. But it makes no difference if you add or subtract 0; you get the same answer. So this problem will have only one answer.

And finally, 3x2+5x+43x2+5x+4 size 12{3x rSup { size 8{2} } +5x+4} {}. Now, b24ac=524(3)(4)=2548=23b24ac=524(3)(4)=2548=23 size 12{b rSup { size 8{2} } - 4 ital "ac"=5 rSup { size 8{2} } - 4 $$3$$ $$4$$ ="25" - "48"= - "23"} {}. So in the numerator we will have 2323 size 12{ sqrt { - "23"} } {}. Since you cannot take the square root of a negative number, there will be no solutions!

Summary: The Discriminant

• If the discriminant is a perfect square, you will have two rational solutions.
• If the discriminant is a positive number that is not a perfect square, you will have two irrational solutions (ie they will have square roots in them).
• If the discriminant is 0, you will have one solution.
• If the discriminant is negative, you will have no solutions.

These rules do not have to be memorized: you can see them very quickly by understanding the quadratic formula (which does have to be memorized—if all else fails, try singing it to the tune of Frère Jacques).

Why is it that quadratic equations can have 2 solutions, 1 solution, or no solutions? This is easy to understand by looking at the following graphs. Remember that in each case the quadratic equation asks when the function is 0—that is to say, when it crosses the xx size 12{x} {}-axis.

More on how to generate these graphs is given below. For the moment, the point is that you can visually see why a quadratic function can equal 0 twice, or one time, or never. It can not equal 0 three or more times.

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.

PDF | EPUB (?)

What is an EPUB file?

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

Collection to:

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

Module to:

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