A Quadratic within a Quadratic
Example
A quadratic within a quadratic
Solve (x^{2} + 2x)^{2}  11(x^{2} + 2x) + 24 = 0.
Solution
Note that x^{2} + 2x and (x^{2} + 2x)^{2 }appear in the equation. Let a
= x^{2} + 2x and
substitute.



a^{2}  11a + 24 
= 0 




(a  8)(a  3) 
= 0 
Factor. 
a  8 
= 0 
or 
a  3 
= 0 

a 
= 8 
or 
a 
= 3 

x^{2} + 2x 
= 8 
or 
x^{2} + 2x 
= 3 
Replace a by x^{2} + 2x. 
x^{2} + 2x  8 
= 0 
or 
x^{2} + 2x  3 
= 0 

(x  2)(x + 4) 
= 0 
or 
(x + 3)(x  1) 
= 0 

x  2 = 0 
or 
x + 4 
= 0 
or 
x + 3 
= 0 
or 
x  1 
= 0 
x = 2 
or 
x 
= 4 
or 
x 
= 3 
or 
x 
= 1 
Check. The solution set is {4, 3, 1, 2}.
