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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Solving Nonlinear Equations by Factoring

In the next example, we will first factor out a common factor.

Example

Solve for w: -w4 + 48w2 = -2w3

 Solution 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  -w4 + 2w3 + 48w2 w4 - 2w3 - 48w2w2[w2 - 2w - 48]w2[(w - 8)(w + 6)] = -2w3  = 0 = 0= 0= 0 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 -6.

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.