1. Equations Reducible to Quadratic
Section 11.5
Equations Reducible to Quadratic
Phong Chau

2. 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 = 0.

3. Example Solve x4 – 5x2 + 4 = 0. Solution u2 – 5u + 4 = 0
Let u = x2. Then we solve by substituting u for x2 and u2 for x4:
u2 – 5u + 4 = 0
(u – 1)(u – 4) = 0
Factoring
Principle of zero products
u – 1 = 0 or u – 4 = 0
u = 1 or u = 4

4. Check: Replace u with x2 TRUE TRUE x2 = 1 or x2 = 4 x = 1: x = 2:
(16) – 5(4) + 4 = 0
(1) – 5(1) + 4 = 0
TRUE
TRUE
The solutions are 1, –1, 2, and –2.

5. Example Solve Solution u2 – 8u – 9 = 0 (u – 9)(u +1) = 0
Let u = Then we solve by substituting u for
and u2 for x:
u2 – 8u – 9 = 0
(u – 9)(u +1) = 0
u – 9 = 0 or u + 1 = 0
u = 9 or u = –1

6. Check:
x = 81:
x = 1:
FALSE
TRUE
The solution is 81.

7. Example Solve Solution u2 + 4u – 2 = 0
Let u = t −1. Then we solve by substituting u for t −1 and u2 for t −2:
u2 + 4u – 2 = 0

9. Examples
Solve the following equations: