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Solving Nonlinear Equations by Factoring

Example

Solve for x: x3 + 2x = 6 + 3x2
Solution

Step 1 Write the equation in standard form.

Subtract 6 and 3x2 from both sides.

x3 + 2x

 

x3 - 3x2 + 2x - 6

= 6 + 3x2

 

= 0

Step 2 Factor by grouping.

Factor out the common factor, (x - 3).

Step 3 Use the Zero Product Property.

x2(x - 3) + 2(x - 3)

(x - 3)(x2 + 2)

x - 3 = 0 or x2 + 2

= 0

= 0

= 0

Step 4 Solve for the variable.

x = 3 or x2

= -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 x3 + 2x = 6 + 3x2 written in standard form is x3 - 3x2 + 2x - 6 = 0. The graph of the corresponding function, f(x) = x3 - 3x2 + 2x - 6 is shown.

The graph crosses the x-axis 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.

 
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