Understanding the Discriminant of the Equation x^2 0

The equation (x^2 0) is a simple yet fundamental example in algebra. This equation can be rewritten in the standard form of a quadratic equation, which is (ax^2 bx c 0). For the equation (x^2 0), the coefficients are:

(a 1) (b 0) (c 0)

The discriminant (D) of a quadratic equation (ax^2 bx c 0) is given by the formula:

[D b^2 - 4ac]

Substituting the values of (a), (b), and (c) into the formula, we get:

[D 0^2 - 4 cdot 1 cdot 0 0 - 0 0]

Thus, the discriminant of the equation (x^2 0) is (0). This means that the quadratic equation has exactly one real root, which is a repeated root. In this case, the root is (x 0).

The Role of the Discriminant

The discriminant of a quadratic equation plays a crucial role in determining the nature of its roots. It is a key part of the quadratic formula, which is given by:

[x frac{-b pm sqrt{b^2 - 4ac}}{2a}]

The discriminant, (D b^2 - 4ac), determines the number and type of roots the quadratic equation has:

If (D > 0), the quadratic equation has two distinct real roots. If (D 0), the quadratic equation has exactly one real root (a repeated root). If (D

When the Discriminant is Zero

In the specific case of the equation (x^2 0), since the discriminant is zero, we can conclude that there is exactly one real root. This root can be found by solving the equation directly:

[x^2 0]

a simple solution gives:

[x 0]

Thus, (x 0) is the only root of the equation.

It is important to note that the discriminant provides valuable information about the roots without actually solving the quadratic equation. In this case, the discriminant being zero clearly indicates a repeated root, which is consistent with the direct solution.

Summary

In conclusion, the discriminant of the equation (x^2 0) is zero, indicating a repeated root at (x 0). The discriminant is a powerful tool in the study of quadratic equations, providing insights into the nature of the roots based on the values of the coefficients.