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Equations Quadratic in Form

In the next example we have a fourth-degree equation that is quadratic in form. Note that the fourth-degree equation has four solutions.

 

Example 1

A fourth-degree equation

Solve x4 - 6x2 + 8 = 0.

Solution

Note that x4 is the square of x2. If we let w = x2, then w2 = x4. Substitute these expressions into the original equation.

x4 - 6x2 + 8 = 0  
w2 - 6w + 8 = 0 Replace x4 by w2 and x2 by w.
(w - 2)(w - 4) = 0 Factor.
w - 2 = 0 or w - 4 = 0  
w = 2 or w = 4  
x2 = 2 or x2 = 4 Substitute x2 for w.
x = ± or x =  2 Even-root property

Check. The solution set is {-2, -, , 2}.

Caution

If you replace x2 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.

 
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