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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:

    1st # 2nd # Added

    1 30 31

    2 15 17

    3 10 13

    5 6 11 So the answer is 𝑥 + 5 𝑥 + 6

  • How about these?

    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:

    1st # 2nd # Added

    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:

    1st # 2nd # Added

    Answer: ( )( )

    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:

    1st # 2nd # Added 1st # 2nd # Added

    Answer: ( )( ) Answer: ( )( )

  • 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?

    How about this one? 𝒚𝟐 + 𝟐𝒚 − 𝟗

    If it isn’t, it’s called “Prime”.

    Create your own!

    1st # 2nd # Added

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

    𝟐𝒙𝟐 − 𝟐𝟐𝒙 + 𝟒𝟖

    1st # 2nd # Added

    Becomes

    𝟐(𝒙𝟐 − 𝟏𝟏𝒙 + 𝟐𝟒)

    Answer: ( )( )

  • 𝟑𝒌𝟐 + 𝟐𝟏𝒌 + 𝟑𝟔

    1st # 2nd # Added

    Answer: ( )( )

    𝟒𝒅𝟐 − 𝟒𝒅 − 𝟐𝟒

    1st # 2nd # Added

    Answer: ( )( ) Answer: ( )( )

    −𝟑𝒙𝟐 + 𝟏𝟖𝒙 − 𝟐𝟕

    1st # 2nd # Added

  • 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.

    1st # 2nd # Added

    –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:

    1st # 2nd # Added

    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

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