Solving Nonlinear Equations by Factoring
Example
Solve for x: x^{3} + 2x = 6
+ 3x^{2}
Solution Step 1 Write the equation in standard form.
Subtract 6 and 3x^{2} from both sides.

x^{3} + 2x
x^{3}  3x^{2} +
2x  6 
=
6 + 3x^{2} = 0 
Step 2 Factor by grouping.
Factor out the common factor, (x  3). Step 3 Use the Zero Product Property. 
x^{2}(x
 3) + 2(x  3) (x  3)(x^{2}
+ 2)
x  3 = 0 or x^{2} + 2 
= 0 = 0
= 0 
Step 4 Solve for the variable.

x = 3 or x^{2} 
= 2 
Take the square root of each side.

x 

Write
as an imaginary number. 
x 

So, the three solutions are x = 3,
Note:
Recall that a negative number under a
square root results in an imaginary
number, which we indicate by using the
letter i. Thus,
The equation x^{3} + 2x = 6
+ 3x^{2} written in standard form is x^{3}  3x^{2} +
2x  6 = 0. The graph of the corresponding function,
f(x) = x^{3}  3x^{2} +
2x  6 is shown.
The graph crosses the xaxis at only one location, x = 3. This is because
the only real number solution is x = 3. In a Cartesian coordinate system,
the x and y axes represent real numbers. Therefore, the imaginary
solutions do not appear on the graph. However, the imaginary solutions
check in the original equation.
