Solving Nonlinear Equations by Factoring
In the next example, we will first factor out a common factor.
Example
Solve for w: w^{4} + 48w^{2} = 2w^{3}
Solution
Step 1 Write the equation in standard form.
Add 2w^{3} to both sides.
Multiply both sides by 1 to make
the first term positive.
Step 2 Factor.
Factor out the GCF, w^{2}.
Factor the trinomial.
Step 3 Use the Zero Product Property. 
w^{4} + 48w^{2}
w^{4}
+ 2w^{3} + 48w^{2}
w^{4}  2w^{3}  48w^{2} w^{2}[w^{2}  2w
 48] w^{2}[(w  8)(w + 6)] 
= 2w^{3}
= 0
= 0 = 0 = 0 
Set each factor equal to 0.
Step 4 Solve for the variable. 
w^{2} = 0 or w  8 = 0 or

w + 6 = 0 

So, there are four solutions: 0 (a solution of multiplicity 2), 8, and
6.
The equation w^{4} + 48w^{2} = 2w^{3} written in standard form is w^{4}
2w^{3}  48w^{2} = 0. The graph of the corresponding function,
f(x) = w^{4} 2w^{3}  48w^{2} is shown.
The graph touches or crosses the waxis at the solutions to the equation:
w= 6, w = 0, and w = 8. 