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 Depdendent Variable

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 Dependent Variable

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

Example

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