Equations Quadratic in Form
In the next example we have a fourthdegree equation that is quadratic in form.
Note that the fourthdegree equation has four solutions.
Example 1
A fourthdegree equation
Solve x^{4}  6x^{2} + 8 = 0.
Solution
Note that x^{4} is the square of x^{2}. If we let w = x^{2}, then w^{2}
= x^{4}. Substitute these expressions
into the original equation.
x^{4}  6x^{2} +
8 
= 0 

w^{2}  6w + 8 
= 0 
Replace x^{4} by w^{2} and x^{2}
by w. 
(w  2)(w  4) 
= 0 
Factor. 
w  2 
= 0 
or 
w  4 
= 0 

w 
= 2 
or 
w 
= 4 

x^{2} 
= 2 
or 
x^{2} 
= 4 
Substitute x^{2} for w. 
x 
= Â±

or 
x 
= 2 
Evenroot property 
Check. The solution set is {2, ,
,
2}.
Caution
If you replace x^{2} by w, do not quit when you find the values
of w. If the variable in the original equation is x, then you must solve for x.
Helpful hint
The fundamental theorem of
algebra says that the number
of solutions to a polynomial
equation is less than or equal
to the degree of the polynomial.
This famous theorem
was proved by Carl Friederich
Gauss when he was a young
man.
