Some equations are not quadratic, but can be turned into quadratic equation by using substitution....
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Transcript of Some equations are not quadratic, but can be turned into quadratic equation by using substitution....
![Page 1: Some equations are not quadratic, but can be turned into quadratic equation by using substitution. Such equations are called Equations in Quadratic Form.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649f125503460f94c2666a/html5/thumbnails/1.jpg)
![Page 2: Some equations are not quadratic, but can be turned into quadratic equation by using substitution. Such equations are called Equations in Quadratic Form.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649f125503460f94c2666a/html5/thumbnails/2.jpg)
Some equations are not quadratic, but can be turned into quadratic equation by using substitution. Such equations are called Equations in Quadratic Form or Reducible to Quadratic.
x4 – 5x2 + 4 = 0
(x2)2 – 5(x2) + 4 = 0
u2 – 5u + 4 = 0.
![Page 3: Some equations are not quadratic, but can be turned into quadratic equation by using substitution. Such equations are called Equations in Quadratic Form.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649f125503460f94c2666a/html5/thumbnails/3.jpg)
Solution
Solve x4 – 5x2 + 4 = 0.
u2 – 5u + 4 = 0
Let u = x2. Then we solve by substituting u for x2 and
u2 for x4:
(u – 1)(u – 4) = 0
u = 1 or u = 4
u – 1 = 0 or u – 4 = 0
Factoring
Principle of zero products
Example
![Page 4: Some equations are not quadratic, but can be turned into quadratic equation by using substitution. Such equations are called Equations in Quadratic Form.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649f125503460f94c2666a/html5/thumbnails/4.jpg)
x2 = 1 or x2 = 41 or 2 x x
Check:x = 1: x = 2:
x4 – 5x2 + 4 = 0
(1) – 5(1) + 4 = 0
x4 – 5x2 + 4 = 0
(16) – 5(4) + 4 = 0
The solutions are 1, –1, 2, and –2.
TRUE TRUE
Replace u with x2
![Page 5: Some equations are not quadratic, but can be turned into quadratic equation by using substitution. Such equations are called Equations in Quadratic Form.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649f125503460f94c2666a/html5/thumbnails/5.jpg)
Solution
Solve 8 9 0.x x
u2 – 8u – 9 = 0
(u – 9)(u +1) = 0
u = 9 or u = –1
u – 9 = 0 or u + 1 = 0
xLet u = . Then we solve by substituting u for
and u2 for x:
x
Example
![Page 6: Some equations are not quadratic, but can be turned into quadratic equation by using substitution. Such equations are called Equations in Quadratic Form.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649f125503460f94c2666a/html5/thumbnails/6.jpg)
81 or 1 x x
Check:x = 81: x = 1:
The solution is 81.
FALSE
TRUE
9 or 1x x
8 9 0x x 8 9 0x x
81 8 81 9 0 81 8(9) 9 0
1 8 1 9 0
1 8 9 0 81 72 9 0
![Page 7: Some equations are not quadratic, but can be turned into quadratic equation by using substitution. Such equations are called Equations in Quadratic Form.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649f125503460f94c2666a/html5/thumbnails/7.jpg)
Solution
Solve 2 14 2 0.t t
u2 + 4u – 2 = 0
Let u = t −1. Then we solve by substituting u for t −1
and u2 for t −2:
24 (4) 4(1)( 2)
2(1)u
4 16 82 6
2u
Example
![Page 8: Some equations are not quadratic, but can be turned into quadratic equation by using substitution. Such equations are called Equations in Quadratic Form.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649f125503460f94c2666a/html5/thumbnails/8.jpg)
1 12 6 or 2 6t t
1 12 6 or 2 6
tt
1 1 or t .
2 6 2 6t
![Page 9: Some equations are not quadratic, but can be turned into quadratic equation by using substitution. Such equations are called Equations in Quadratic Form.](https://reader036.fdocuments.us/reader036/viewer/2022082818/56649f125503460f94c2666a/html5/thumbnails/9.jpg)
Examples
Solve the following equations:
6 3
4 2
2
2 2 2
1) 26 27 0
2) 5 4
3 33) 9 6 1 0
2 2
4) ( ) 8( ) 12
x x
x x
x x
x x x x