Equations Quadratic in Form
The next example involves a fractional exponent. To identify this type of
equation as quadratic in form, recall how to square an expression with a fractional
exponent. For example, (x^{1/2})^{2} = x, (x^{1/4})^{2}
= x^{1/2}, and (x^{1/3})^{2} = x^{2/3}.
Example
A fractional exponent
Solve x  9x^{1/2} + 14 = 0.
^{Solution }
Note that the square of x^{1/2} is x. Let w = x^{1/2}; then w^{2}
= (x^{1/2})^{2} = x. Now substitute
w and w^{2} into the original equation:


w^{2}  9w + 14 
= 0 



(w  7)(w  2) 
= 0 

w  7 
= 0 
or 
w  2 
= 0 

w 
= 7 
or 
w 
= 2 

x^{1/2} 
= 7 
or 
x^{1/2} 
= 2 
Replace w by x^{1/2}. 
x 
= 49 
or 
x 
= 4 
Square each side. 
Because we squared each side, we must check for extraneous roots. First evaluate x
 9x^{1/2} + 14 for x = 49:
49  9 Â· 49^{1/2} + 14 = 49  9 Â· 7 + 14 = 0
Now evaluate x  9x^{1/2} + 14 for x = 4:
4  9 Â· 4^{1/2} + 14 = 4  9 Â· 2 + 14 = 0
Because each solution checks, the solution set is {4, 49}.
Caution
An equation of quadratic form must have a term that is the
square of another. Equations such as x^{4}  5x^{3} + 6 = 0 or x^{1/2}
 3x^{1/3}  18 = 0
are not quadratic in form and cannot be solved by substitution.
