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• Module 21.1 – Part 1

Solving Equations By Factoring 𝒙𝟐 + 𝒃𝒙 + 𝒄

How can you use factoring to solve quadratic equations in standard form for which a = 1?

P. 985

• 𝑥 + 1 𝑥 + 3 = 𝑥2 + 3𝑥 + 1𝑥 + 3 = 𝑥2 + 4𝑥 + 3

Here are some quadratic expressions, expanded with FOIL and combining like terms:

𝑥 + 2 𝑥 + 3 = 𝑥2 + 3𝑥 + 2𝑥 + 6 = 𝑥2 + 5𝑥 + 6

𝑥 + 2 𝑥 + 4 = 𝑥2 + 4𝑥 + 2𝑥 + 8 = 𝑥2 + 6𝑥 + 8

𝑥 + 3 𝑥 + 3 = 𝑥2 + 3𝑥 + 3𝑥 + 9 = 𝑥2 + 6𝑥 + 9

𝑥 + 3 𝑥 + 4 = 𝑥2 + 4𝑥 + 3𝑥 + 12 = 𝑥2 + 7𝑥 + 12

These two added together (OI) produce

These two multiplied together (L) produce

F O I L

These two multiplied together (F) produce

• 𝒙𝟐 + 𝟒𝒙 + 𝟑

If I want to factor into its original form, I must find the two numbers that: * When multiplied, produce 3 (at the end) * When added, produce 4 (for the 𝟒𝑥 in the middle)

This one is easy: The numbers are 1 and 3. 1 + 3 = 4 1 * 3 = 3

Producing 𝑥 + 1 𝑥 + 3

• 𝒙𝟐 + 𝟓𝒙 + 𝟔

I must find the two numbers that: * When multiplied, produce 6 (at the end) * When added, produce 5 (for the 𝟓𝑥 in the middle)

They are 2 and 3. Producing 𝑥 + 2 𝑥 + 3

𝒙𝟐 + 𝟔𝒙 + 𝟖

If I want to factor into its original form, I must find the two numbers that: * When multiplied, produce 8 (at the end) * When added, produce 6 (for the 𝟔𝑥 in the middle)

They are ? Producing 𝑥 + 𝑥 +

• 𝒙𝟐 + 𝟏𝟏𝒙 + 𝟑𝟎

I must find the two numbers that when multiplied, produce 30, and when added, produce 11. Let’s work off the multiplication. Here are possibilities:

1 30 31

2 15 17

3 10 13

5 6 11 So the answer is 𝑥 + 5 𝑥 + 6

𝒙𝟐 + 𝟏𝟐𝒙 + 𝟐𝟎

𝒚𝟐 + 𝟗𝒚 + 𝟏𝟖

𝒅𝟐 + 𝟏𝟑𝒅 + 𝟒𝟎

• What if the middle or last term are negative?

𝒏𝟐 + 𝟐𝒏 − 𝟖

I must find the two numbers that: * When multiplied, produce –8 * When added, produce 2

Here are possibilities:

1 –8 –7

–1 8 7

2 –4 –2

–2 4 2 So the answer is 𝑛 − 2 𝑛 + 4

• 𝒙𝟐 + 𝒙 − 𝟏𝟐 I must find the two numbers that: * When multiplied, produce ? * When added, produce ? Here are possibilities:

𝒄𝟐 − 𝟓𝒄 − 𝟐𝟒 𝒙𝟐 − 𝟗𝒙 + 𝟏𝟖 I must find the two numbers that: * When multiplied, produce ? * When added, produce ? Here are possibilities:

I must find the two numbers that: * When multiplied, produce ? * When added, produce ? Here are possibilities:

• Signs Of Factors

Sign

of b

Sign

of c Example

Sign of

Factor 1

Sign of

Factor 2

Example

Factored Note

+ + 𝒙𝟐 + 𝟔𝒙 + 𝟖 + + (𝒙 + 𝟒)(𝒙 + 𝟐)

– – 𝒙𝟐 − 𝟐𝒙 − 𝟏𝟓 + – (𝒙 + 𝟑)(𝒙 − 𝟓) Factor with greater AV is – (5)

+ – 𝒙𝟐 + 𝟐𝒙 − 𝟖 + – (𝒙 + 𝟒)(𝒙 − 𝟐) Factor with greater AV is + (4)

– + 𝒙𝟐 − 𝟖𝒙 + 𝟏𝟐 – – (𝒙 − 𝟔)(𝒙 − 𝟐)

• Is every quadratic function factorable?

If it isn’t, it’s called “Prime”.

• Sometimes you can remove a GCF first, which will make factoring easier.

𝟐𝒙𝟐 − 𝟐𝟐𝒙 + 𝟒𝟖

Becomes

𝟐(𝒙𝟐 − 𝟏𝟏𝒙 + 𝟐𝟒)

• 𝟑𝒌𝟐 + 𝟐𝟏𝒌 + 𝟑𝟔

𝟒𝒅𝟐 − 𝟒𝒅 − 𝟐𝟒

−𝟑𝒙𝟐 + 𝟏𝟖𝒙 − 𝟐𝟕

• P. 989

Look at a previous slide: The factors will have the same sign and both will be negative.

• 𝒙𝟐 − 𝟖𝒙 + 𝟏𝟐 = 𝟎

I must find the two numbers that when multiplied, produce 12, and when added, produce –8. Here are possibilities, keeping in mind that both numbers will have the same sign and both will be negative.

–1 –12 –13

–2 –6 –8

–3 –4 –7

So the answer is 𝑥 − 2 𝑥 − 6 = 0

Using the Zero Product Property 𝑥 = 2 𝑥 = 6

These are the Solutions aka X-intercepts aka Zeros aka Roots

P. 989

• The “related” function is 𝒇 𝒙 = 𝒙𝟐 − 𝟖𝒙 + 𝟏𝟐 Here’s its graph:

Note the x-intercepts are x = 2 and x = 6.

P. 989

• P. 989

I must find the two numbers that when multiplied, produce –15, and when added, produce –2. Here are possibilities:

So the answer is 𝑥 𝑥 = 0

Using the Zero Product Property 𝑥 = 𝑥 =

These are the Solutions aka X-intercepts aka Zeros aka Roots

• P. 989The “related” function is 𝒇 𝒙 = 𝒙𝟐 − 𝟐𝒙 − 𝟏𝟓

Here’s its graph. Note the x-intercepts are x = –3 and x = 5.

P. 990

• P. 991

• P. 994