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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.