Solving Nonlinear Equations by Factoring
Solve for x: x^{4}  21x^{2}  35 = 65
Example
Solution
Step 1 Write the equation in standard form.
Subtract 65 from both sides.
Step 2 Factor.
Step 3 Use the Zero Product Property.
Step 4 Solve for the variable. 
x^{4}  21x^{2}  35
x^{4}  21x^{2}  100 (x^{2}
+ 4)(x^{2}  25) x^{2} + 4 = 0 or x^{2}
25 x^{2} =  4 or x^{2} 
= 65
= 0 = 0 = 0 = 25 


So, there are four solutions: 2i, +2i, 5, and +5.
The equation x^{4}  21x^{2}  35 = 65 written in standard form is x^{4}  21x^{2}  100
= 0. The graph of the corresponding function,
f(x) = x^{4}  21x^{2}  100 is shown.
The graph crosses the xaxis at only two locations, x = 5 and x
= 5. This is because these are the only real number solutions.
