Solving Nonlinear Equations by Factoring
In the next example, we will first factor out a common factor.
Solve for w: -w4 + 48w2 = -2w3
Step 1 Write the equation in standard form.
Add 2w3 to both sides.
Multiply both sides by -1 to make
the first term positive.
Step 2 Factor.
Factor out the GCF, w2.
Factor the trinomial.
Step 3 Use the Zero Product Property.
|-w4 + 48w2
+ 2w3 + 48w2
w4 - 2w3 - 48w2
w2[w2 - 2w
w2[(w - 8)(w + 6)]
|Set each factor equal to 0.
Step 4 Solve for the variable.
|w2 = 0 or w - 8 = 0 or
|w + 6 = 0
So, there are four solutions: 0 (a solution of multiplicity 2), 8, and
The equation -w4 + 48w2 = -2w3 written in standard form is w4
-2w3 - 48w2 = 0. The graph of the corresponding function,
f(x) = w4 -2w3 - 48w2 is shown.
The graph touches or crosses the w-axis at the solutions to the equation:
w= -6, w = 0, and w = 8.