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.

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

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

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

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

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

1 12 6 or 2 6t t

1 12 6 or 2 6

tt

1 1 or t .

2 6 2 6t

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