91Ó°ÊÓ

Factoring is a fundamental algebraic technique used to simplify and solve polynomial equations. In our original problem, the equation is given as \(x^4 + 4x^3 + 2x^2 = 0\). The first step in solving this is to identify any common factors. Here, \(x^2\) is common in all terms. By factoring \(x^2\) out of the equation, we simplify it to \(x^2(x^2 + 4x + 2) = 0\). This step is crucial because it breaks down a complex fourth-degree polynomial into a much simpler form, making it easier to find solutions.

* **Why is factoring useful?** It's useful because it transforms equations into smaller, manageable parts. This leads directly to finding solutions by setting each factor equal to zero.
* **What do we do after factoring?** We solve each of the resulting simpler equations. In this case, \(x^2 = 0\) gives a double root of \(x = 0\). The other factor, \(x^2 + 4x + 2 = 0\), is further solved using the quadratic formula.

The quadratic formula is a handy tool when you need to find solutions to quadratic equations that can't be easily factored. In the equation \(x^2 + 4x + 2 = 0\), we can't factor it easily into integers, so we use the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• "\(a\)", "\(b\)", and "\(c\)" are coefficients from the equation \(ax^2 + bx + c = 0\)

For our equation, \(a = 1\), \(b = 4\), and \(c = 2\), which we substitute into the formula. This gives us solutions based on the identified discriminant. Using the formula helps in finding both the real and complex solutions of any quadratic equation.

Real solutions of an equation are the values of \(x\) that satisfy the equation and are real (non-imaginary) numbers. In the context of the exercise, once we factor and apply the quadratic formula, we find the real roots. After solving \(x^2 = 0\), we obtain \(x = 0\), which is a real solution known as a double root because it occurs twice.

Then, we delve into the quadratic formula for \(x^2 + 4x + 2 = 0\). After calculating with the discriminant, we find two additional real solutions: \(x = -2 + \sqrt{2}\) and \(x = -2 - \sqrt{2}\). All these solutions are real because they are not accompanied by the imaginary unit \(i\). In problems like these, real solutions tell us where a polynomial function intersects the x-axis when graphed.

The discriminant in the quadratic formula, given by \(b^2 - 4ac\), is crucial as it informs us about the nature of the solutions of the quadratic equation.

* **Positive Discriminant:** Indicates two distinct real solutions. For example, in our exercise, the discriminant is \(16 - 8 = 8\), which is positive.
* **Zero Discriminant:** Indicates one real solution, sometimes called a repeated or double root.
* **Negative Discriminant:** Suggests the existence of two complex solutions. These solutions include the imaginary unit \(i\).

In this exercise, since the discriminant is positive, it confirms the existence of real solutions \(x = -2 + \sqrt{2}\) and \(x = -2 - \sqrt{2}\). Understanding the discriminant helps predict the type of solutions before actually solving the equation, saving time and providing conceptual clarity.