![Page 1: Derivation of the Quadratic Formula The following shows how the method of Completing the Square can be used to derive the Quadratic Formula. Start with](https://reader036.vdocuments.net/reader036/viewer/2022083008/56649f455503460f94c663af/html5/thumbnails/1.jpg)
Derivation of the Quadratic Formula
• The following shows how the method of Completing the Square can be used to derive the Quadratic Formula. Start with the general
quadratic equation.
Move the constant to the other side.
2 0ax bx c
Divide by a, the leading coefficient.
2ax bx c
2 b cx x
a a
![Page 2: Derivation of the Quadratic Formula The following shows how the method of Completing the Square can be used to derive the Quadratic Formula. Start with](https://reader036.vdocuments.net/reader036/viewer/2022083008/56649f455503460f94c663af/html5/thumbnails/2.jpg)
Complete the square.
Add the value to both sides of the equation.
2 b cx x
a a
1
2 2
b b
a a
2 2
22 4
b b
a a
2 22
2 24 4
b b b cx x
a a a a
![Page 3: Derivation of the Quadratic Formula The following shows how the method of Completing the Square can be used to derive the Quadratic Formula. Start with](https://reader036.vdocuments.net/reader036/viewer/2022083008/56649f455503460f94c663af/html5/thumbnails/3.jpg)
Factor the left side and simplify the right side.
2 22
2 24 4
b b b cx x
a a a a
2 2
2
4
2 4 4
b b c ax
a a a a
2
2 2
4
4 4
b ac
a a
2
2
4
4
b ac
a
![Page 4: Derivation of the Quadratic Formula The following shows how the method of Completing the Square can be used to derive the Quadratic Formula. Start with](https://reader036.vdocuments.net/reader036/viewer/2022083008/56649f455503460f94c663af/html5/thumbnails/4.jpg)
Complete the problem using the square root method of solving a quadratic equation.
2 2
2
4
2 4
b b acx
a a
2
2
4
2 4
b b acx
a a
2 4
2 2
b b acx
a a
![Page 5: Derivation of the Quadratic Formula The following shows how the method of Completing the Square can be used to derive the Quadratic Formula. Start with](https://reader036.vdocuments.net/reader036/viewer/2022083008/56649f455503460f94c663af/html5/thumbnails/5.jpg)
2 4
2 2
b b acx
a a
2 4
2 2
b b acx
a a
2 4
2
b b acx
a
![Page 6: Derivation of the Quadratic Formula The following shows how the method of Completing the Square can be used to derive the Quadratic Formula. Start with](https://reader036.vdocuments.net/reader036/viewer/2022083008/56649f455503460f94c663af/html5/thumbnails/6.jpg)
Quadratic Formula
2 4
2
b b acx
a
The expression under the radical is called the Discriminant.
2 4b ac
![Page 7: Derivation of the Quadratic Formula The following shows how the method of Completing the Square can be used to derive the Quadratic Formula. Start with](https://reader036.vdocuments.net/reader036/viewer/2022083008/56649f455503460f94c663af/html5/thumbnails/7.jpg)