How can I solve the equation x^3 + 2x^2 + 0 using the substitution u = x^2, and what are the steps to factor it?

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Solving the Equation

To solve the equation x³ + 2x² + 0 = 0, we start by substituting u = x². This simplifies the equation significantly.

Step 1: Rewrite in Terms of u

Substituting u into the equation, we get:

u² + 2u + 0 = 0

Step 2: Factor the Equation

Next, we need to factor the quadratic equation u² + 2u = 0.

By factoring, we can express it as:

u(u + 2) = 0

Step 3: Find the Solutions for u

The factored equation u(u + 2) = 0 gives us the solutions:

  • u = 0
  • u + 2 = 0 → u = -2

Step 4: Return to x

Now we substitute back for x. Recall that u = x²:

  • x² = 0 → x = 0
  • x² = -2 → x = ±√-2 → x = ±i√2

Final Solutions

Thus, the solutions to the original equation x³ + 2x² = 0 are:

  • x = 0
  • x = ±i√2 (the imaginary solutions)

In summary, we used a substitution method to simplify and factor the equation, and by doing so, we identified both real and imaginary solutions for x.

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